EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2022-198 LHCb-PAPER-2022-031 October 19, 2022

Template for writing LHCb papers

LHCb Collaboration

An amplitude analysis of
${{{\mathrm{B}}^{-}}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}}$
decays is performed using about 4400 signal candidates selected on a data
sample of ${\mathrm{p}}{\mathrm{p}}$ collisions recorded at center-of-mass
energies of 7, 8 and 13  TeV with the LHCb detector, corresponding to an
integrated luminosity of 9 $\text{\,fb}^{-1}$. A narrow resonance in the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ $\Lambda$ system, consistent
with a pentaquark candidate with strangeness, is observed with high
significance. The mass and the width of this new state are measured to be
$4338.2\pm 0.7\pm 0.4\text{\,Me\kern-1.00006ptV}$ and ${7.0\pm 1.2\pm
1.3\text{\,Me\kern-1.00006ptV}}$, where the first uncertainty is statistical
and the second systematic. The spin is determined to be $1/2$ and negative
parity is preferred. Due to the small $Q$-value of the reaction, the most
precise single measurement of the ${{\mathrm{B}}^{-}}$ mass to date,
$5279.44\pm 0.05\pm 0.07\text{\,Me\kern-1.00006ptV}$, is obtained.

Submitted to Phys. Rev. Lett.

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

The discovery of pentaquark candidates in the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ $\mathrm{p}$ system at LHCb
[LHCb-PAPER-2015-029, LHCb-PAPER-2019-014] opened a new field of investigation
in baryon spectroscopy. Such resonant structures with valence quark
content111The exotic hadron naming scheme defined in Ref. [Gershon:2814506] is
used throughout this Letter. ${P_{\psi}^{N}}^{+}$ = $\mathrm{c}$
$\overline{{\mathrm{c}}}$$\mathrm{u}$ $\mathrm{u}$ $\mathrm{d}$ have been
observed only in the
${{\Lambda}^{0}_{\mathrm{b}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\mathrm{p}}{{\mathrm{K}}^{-}}$
decay to date. Recently, evidence for a new ${P_{\psi}^{N}}^{+}$= $\mathrm{c}$
$\overline{{\mathrm{c}}}$$\mathrm{u}$ $\mathrm{u}$ $\mathrm{d}$ candidate
was found in the
${{\mathrm{B}}^{0}_{\mathrm{s}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\mathrm{p}}{\overline{\mathrm{p}}}$
decay [LHCb-PAPER-2018-046, LHCb-PAPER-2021-018], and evidence for a ${P_{\psi
s}^{{\Lambda}}}^{0}$ = $\mathrm{c}$ $\overline{{\mathrm{c}}}$$\mathrm{u}$
$\mathrm{d}$ $\mathrm{s}$ pentaquark candidate with strangeness was found in
the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ $\Lambda$ system in the
${{\Xi}^{-}_{\mathrm{b}}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{{\mathrm{K}}^{-}}$
decay [LHCb-PAPER-2020-039]222Charge conjugation is implied throughout this
Letter..

Pentaquarks are predicted within the quark model to have a minimal quark
content of three quarks plus a quark-antiquark pair. Experimentally, the
pentaquark candidates are found close to threshold for the production of
ordinary baryon-meson states, i.e. $\varSigma_{c}^{+}\overline{D}^{0}$ and
$\varSigma_{c}^{+}\overline{D}^{*0}$ for the observed ${P_{\psi}^{N}}^{+}$
states [LHCb-PAPER-2015-029, LHCb-PAPER-2019-014], and
${\varXi}_{c}^{0}\overline{D}^{*0}$ for the ${P_{\psi s}^{{\Lambda}}}^{0}$
state [LHCb-PAPER-2020-039]. Various interpretations of these states have been
proposed, including tightly bound pentaquark states [Esposito:2016noz,
Richard:2016eis, Dong:2021juy], loosely bound baryon-meson molecular states
([Guo:2017jvc] and references therein), and rescattering effects
[Guo:2019twa]. Hidden-charm pentaquarks with strangeness were predicted in
[Xiao:2019gjd, Wang:2019nvm] as hadronic molecules, and in [Ali:2019clg] as
compact states. However, their nature is still largely unknown and further
investigation is needed [Olsen:2017bmm].

The
${{{\mathrm{B}}^{-}}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}}$
decay offers the unique opportunity to simultaneously search for
${\overline{P}_{\psi}^{N}}^{-}$ and ${P_{\psi s}^{{\Lambda}}}^{0}$ pentaquark
candidates in the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$
$\overline{\mathrm{p}}$ and ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$
$\Lambda$ systems, respectively. In particular, the phase space available in
the decay allows searches for pentaquark candidates located close to different
baryon-meson thresholds, such as ${\Lambda}^{+}_{\mathrm{c}}$
${\mathrm{D}}^{0}$ for ${P_{\psi}^{N}}^{+}$, and ${\Lambda}^{+}_{\mathrm{c}}$
${\mathrm{D}}^{-}_{\mathrm{s}}$, ${\Xi}^{+}_{\mathrm{c}}$ ${\mathrm{D}}^{-}$
for ${P_{\psi s}^{{\Lambda}}}^{0}$ candidates. Neither the ${P_{\psi
s}^{{\Lambda}}(4459)}^{0}$ state, found in the
${{\Xi}^{-}_{\mathrm{b}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{{\mathrm{K}}^{-}}$
decay [LHCb-PAPER-2020-039], nor the ${P_{\psi}^{N}(4337)}^{+}$ state, found
in the
${{\mathrm{B}}^{0}_{\mathrm{s}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\mathrm{p}}{\overline{\mathrm{p}}}$
decays [LHCb-PAPER-2021-018], is accessible with the present analysis since
they are outside of the available phase space.

The small $Q$-value of the decay, approximately333Natural units with
$\hslash=c=1$ are used throughout this Letter.
$128\text{\,Me\kern-1.00006ptV}$, provides excellent mass resolution, allowing
searches for narrow resonant structures. This decay was previously studied by
the CMS collaboration using a sample of $450\pm 20$ signal candidates and the
invariant mass distributions of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ $\Lambda$,
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ $\overline{\mathrm{p}}$,
$\Lambda$ $\overline{\mathrm{p}}$ systems were found to be inconsistent with
the pure phase-space hypothesis [CMS:2019kbn]. In this Letter, an amplitude
analysis of the
${{{\mathrm{B}}^{-}}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}}$
decay is performed using signal candidates selected on a data sample of
${\mathrm{p}}{\mathrm{p}}$ collisions at centre-of-mass energies of 7 TeV and
8 TeV (Run 1), and 13 TeV (Run 2), recorded between 2011 and 2018 by the LHCb
detector, corresponding to an integrated luminosity of 9 $\text{\,fb}^{-1}$.
In the following, the first observation of a ${P_{\psi s}^{{\Lambda}}}^{0}$
pentaquark candidate with strangeness in the
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}$ system is reported,
which is different from the ${P_{\psi s}^{{\Lambda}}(4459)}^{0}$ state found
in the
${{\Xi}^{-}_{\mathrm{b}}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{{\mathrm{K}}^{-}}$
decay [LHCb-PAPER-2020-039].

The LHCb detector is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, described in detail in Refs. [LHCb-
DP-2014-002, LHCb-DP-2014-001, LHCb-DP-2013-003, LHCb-DP-2012-002]. The online
event selection is performed by a trigger [LHCb-DP-2012-004], comprising a
hardware stage based on information from the muon system which selects
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}\rightarrow{\upmu^{+}}{\upmu^{-}}$
decays, followed by a software stage that applies a full event reconstruction.
The software trigger relies on identifying
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ decays into muon pairs
consistent with originating from a $\mathrm{B}$-meson decay vertex detached
from the primary $\mathrm{p}$ $\mathrm{p}$ collision point.

Samples of simulated events are used to study the properties of the signal
mode decay
${{{{\mathrm{B}}^{-}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}(\rightarrow{\mathrm{p}}{{\uppi}^{-}}){\overline{\mathrm{p}}}}$
and the control mode decay
${{{{\mathrm{B}}^{-}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{{\mathrm{K}}^{*-}}(\rightarrow{{\mathrm{K}}^{0}_{\mathrm{S}}}(\rightarrow{{\uppi}^{+}}{{\uppi}^{-}}){{\uppi}^{-}})}$
. The latter are used to calibrate the distributions of simulated
${{\mathrm{B}}^{-}}$ decays with data.

The $\mathrm{p}$ $\mathrm{p}$ collisions are generated using Pythia
[Sjostrand:2007gs] with a specific LHCb configuration [LHCb-PROC-2010-056].
Decays of hadronic particles and interactions with the detector material are
described by EvtGen [Lange:2001uf], using Photos [davidson2015photos], and by
the Geant4 toolkit [Allison:2006ve, *Agostinelli:2002hh, LHCb-PROC-2011-006],
respectively. The signal and the control mode decays are generated from a
uniform phase-space distribution.

Signal ${{\mathrm{B}}^{-}}$ candidates are formed from combinations of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$, $\Lambda$ and
$\overline{\mathrm{p}}$ candidates originating from a common decay vertex. The
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ candidates are formed from pairs
of oppositely charged tracks identified as muons and originating from a decay
vertex significantly displaced from the associated $\mathrm{p}$ $\mathrm{p}$
primary vertex (PV). The associated PV for a given particle is the PV with the
smallest impact parameter $\chi^{2}_{\rm IP}$, defined as the difference in
the vertex-fit $\chi^{2}$ of a given PV reconstructed with and without the
particle under consideration. The
${{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}$ candidates are formed from
pairs of oppositely charged tracks and selected in two different categories
according to the $\Lambda$ decay position: i) the “long” category for early
decays that allow the proton and pion candidates to be reconstructed in the
vertex detector; ii) the “downstream” category for $\Lambda$ baryons that
decay outside the vertex detector and are reconstructed in the tracking
stations only. The long candidates have better mass, momentum and vertex
resolution than downstream candidates. The $\overline{\mathrm{p}}$ candidate
is a charged track identified as an antiproton.

A kinematic fit [Hulsbergen:2005pu] to the ${{\mathrm{B}}^{-}}$ candidate is
performed with the dimuon and the $\mathrm{p}$ ${\uppi}^{-}$ masses
constrained to the known ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ and
$\Lambda$ masses, respectively [PDG2022]. Simulated events are weighted such
that the distributions of transverse momentum, $p_{\mathrm{T}}$, and number of
tracks per event for ${{\mathrm{B}}^{-}}$ candidates match the
${{{\mathrm{B}}^{-}}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{{\mathrm{K}}^{*-}}$
control-mode distributions in data. In simulation the particle identification
(PID) variables for each charged track are resampled as a function of their
$p$, $p_{\mathrm{T}}$ and the number of tracks in the event using
${{\Lambda}^{+}_{\mathrm{c}}}\\!\rightarrow{\mathrm{p}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$
and $D^{\ast+}\rightarrow D^{0}(\rightarrow K^{-}{{\uppi}^{+}})\pi^{+}$
calibration samples from data [LHCb-DP-2018-001].

The final stage of the selection uses multivariate techniques trained with
simulation and data. Separate boosted-decision-tree (BDT [Breiman])
classifiers are employed for the four combinations of two data taking periods
(Run 1 and Run 2) and two signal categories, using long and downstream
reconstructed $\Lambda$ candidates. Each BDT is trained on simulated signal
decays and data sidebands, with the
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}})$
invariant mass in the range $[5320,5360]\text{\,Me\kern-1.00006ptV}$. The
variables used as input to the BDT are: the $p_{\mathrm{T}}$, the decay length
significance, the angle between the momentum and the flight direction and the
$\chi^{2}_{\rm IP}$ variable of the ${{\mathrm{B}}^{-}}$ candidate; the
$\chi^{2}$ probability from the kinematic fit of the candidate; the sum of the
$\chi^{2}_{\rm IP}$ of the daughter particles; the angle between the momentum
and the flight direction, the $\chi^{2}$ of the flight distance (only for long
category candidates), the $\chi^{2}_{\rm IP}$ variables of the $\Lambda$
candidate; and the hadron PID for the $\overline{\mathrm{p}}$ candidate from
the ring-imaging Cherenkov detectors.

The BDT output selection criterion is chosen by maximising the figure of merit
$\mathcal{S}^{2}/(\mathcal{S}+\mathcal{B})^{3/2}$, where $\mathcal{S}$ and
$\mathcal{B}$ are the signal and background yield in a region of $\pm
5.3\text{\,Me\kern-1.00006ptV}$ around the known ${{\mathrm{B}}^{-}}$ mass. To
avoid a possible bias due to fluctuations of the signal yield, $\mathcal{S}$
is determined from a fit to the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$
$\Lambda$ $\overline{\mathrm{p}}$ invariant-mass distribution in data after
applying a loose BDT selection, multiplied by the efficiency of the BDT output
requirement obtained from simulation. Similarly, $\mathcal{B}$ is extracted
from a fit to sideband data.

For candidates passing all selection criteria, a maximum-likelihood fit is
performed to the
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}})$
distribution shown in Fig. 1, resulting in a signal yield of $4620\pm 70$. For
the amplitude analysis about 4400 signal candidates are retained, with a
purity of $93.0\%$ in the signal region of $\pm 2.5\sigma$ around the mass
peak, where $\sigma\approx 2.1\text{\,Me\kern-1.00006ptV}$ is the mass
resolution. The signal distribution is modelled by the sum of a Johnson
function [JohnsonSU] and two Crystal Ball [Skwarnicki:1986xj] functions
sharing the same mean and width parameters determined from the fit. The tail
parameters and fractions of each signal component are fixed to values obtained
from a fit to simulated events. The background contribution is mainly due to
random combinations of charged particles in the event and is described by a
third-order Chebyshev polynomial.

Figure 1: Invariant mass distribution of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ $\Lambda$
$\overline{\mathrm{p}}$ candidates. The data are overlaid with the results of
the fit.

The Dalitz distribution of the reconstructed ${{\mathrm{B}}^{-}}$ candidates
in the signal region is shown in Fig. 2, where a horizontal band in the region
around $18.8\text{\,Ge\kern-1.00006ptV}^{2}$ in the
$m^{2}({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda})$ distribution
is present. Some structure in the high
$m^{2}({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}})$
spectrum is also present. This Letter investigates the nature of these
enhancements.

Figure 2: Dalitz distribution for ${{\mathrm{B}}^{-}}$ candidates in the
signal region.

An amplitude analysis of the ${{\mathrm{B}}^{-}}$ candidates in the signal
region is performed using a phenomenological model based on the interference
of two-body resonances in the three decay chains,
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}K^{*-}(\rightarrow{\Lambda}{\overline{\mathrm{p}}})$,
${\Lambda}\overline{P}_{\psi}^{N-}(\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}})$,
and ${\overline{\mathrm{p}}}{P_{\psi
s}^{{\Lambda}}}^{0}(\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda})$,
labelled as the $K^{*-}$, ${\overline{P}_{\psi}^{N}}^{-}$ and ${P_{\psi
s}^{{\Lambda}}}^{0}$ chains, respectively. The angular information of the
subsequent
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}\\!\rightarrow{\upmu^{+}\upmu^{-}}$
and ${\Lambda}\\!\rightarrow{\mathrm{p}}{{\uppi}^{-}}$ decays are taken into
account in all cases. The decay amplitudes are based on helicity formalism
[Chung:186421] with $C\\!P$ symmetry enforced, and follow the prescriptions in
Ref.[DPdeco] for the spin alignment of the different decay chains. Details
about the decay amplitude definition are given in the Supplemental material
[SupplementalMaterial].

The decay amplitudes are defined as a function of the six-dimensional phase
space of the ${{\mathrm{B}}^{-}}$ decay,
$(m_{{\Lambda}{\overline{\mathrm{p}}}},\vec{\Omega})$, described by the
combined invariant mass $m_{{\Lambda}{\overline{\mathrm{p}}}}$ of the
$\overline{\mathrm{p}}$ and $\Lambda$ pairs, and by five angular variables
indicated as $\vec{\Omega}$: the cosine of the helicity angle,
$\cos\theta_{{{\mathrm{K}}^{*}}}$
$(\cos\theta_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}})$, of the
$\Lambda$ $({\upmu^{-}})$ in the $\Lambda$ $\overline{\mathrm{p}}$
(${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$) rest frame; the azimuthal
angle, $\phi_{\mathrm{p}}$ $(\phi_{\upmu^{-}})$, of the ${\mathrm{p}}$
(${\upmu^{-}}$) in the rest frame of the $\Lambda$
(${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$); and the cosine of the
helicity angle, $\cos\theta_{{\Lambda}}$, of the $\mathrm{p}$ in the rest
frame of the $\Lambda$. The amplitude fit to determine the model parameters
$\vec{\omega}$, i.e. the couplings, the masses, the widths, and lineshape
parameters of different contributions, is performed by minimising the negative
log-likelihood function,

$\displaystyle-2\log{{\cal
L}}(\vec{\omega})=-2\sum_{i}\log{\left[(1-\beta){\cal
P}_{\rm{sig}}(m_{{\Lambda}{\overline{\mathrm{p}}},i},\vec{\Omega_{i}}|\vec{\omega})+\beta{\cal
P_{\rm{bkg}}}(m_{{\Lambda}{\overline{\mathrm{p}}},i},\vec{\Omega}_{i})\right]},$
(1)

where ${\cal P}_{\rm{sig}}$ $({\cal P}_{\rm{bkg}})$ is the probability density
function (PDF) for the signal (background) component of the $i$th event, and
$\beta=0.07\pm 0.01$ is the fraction of background candidates in the signal
region. The signal PDF is proportional to the squared decay amplitude $|{\cal
M}(m_{{\Lambda}{\overline{\mathrm{p}}}},\vec{\Omega}|\vec{\omega})|^{2}$, and
accounts for the phase-space element
$\Phi(m_{{\Lambda}{\overline{\mathrm{p}}}})$ and the reconstruction efficiency
$\epsilon(m_{{\Lambda}{\overline{\mathrm{p}}}},\vec{\Omega})$,

${\cal
P}_{\rm{sig}}(m_{{\Lambda}{\overline{\mathrm{p}}}},\vec{\Omega}|\vec{\omega})=\frac{|{\cal
M}(m_{{\Lambda}{\overline{\mathrm{p}}}},\vec{\Omega}|\vec{\omega})|^{2}\Phi(m_{{\Lambda}{\overline{\mathrm{p}}}})\epsilon(m_{{\Lambda}{\overline{\mathrm{p}}}},\vec{\Omega})}{I(\vec{\omega})}.$
(2)

The denominator, $I(\vec{\omega})$, normalizes the probability. The background
PDF, ${\cal P}_{\rm bkg}$, is parameterized according to a six-dimensional
phase-space function based on Legendre polynomials, whose coefficients are
determined from the
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}})$
region ${[5200,5250]\cup[5340,5350]\text{\,Me\kern-1.00006ptV}}$. Similarly,
the reconstruction efficiency is parameterized using Legendre polynomials with
coefficients determined using simulated phase-space signal decays.

No well-established resonances are expected to decay into the
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}$ and
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}}$ final
states. However, excited $K^{*-}$ resonances decaying outside of the phase
space of the
${{{{\mathrm{B}}^{-}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}}}$
decay can contribute to the ${\Lambda}{\overline{\mathrm{p}}}$ channel
[CMS:2019kbn]. A fit including only NR contributions and
$K^{*}_{4}(2045)^{-}$, $K^{*}_{2}(2250)^{-}$ and $K^{*}_{3}(2320)^{-}$
resonant amplitudes does not reproduce the data distribution. A $\chi^{2}/{\rm
n.d.f.}$ of $123.2/46$ is obtained, where the $\chi^{2}$ is calculated as the
largest value over the six one-dimensional fit projections and the ${\rm
n.d.f.}$ is extracted from pseudoexperiments by fitting the tail of the
$\chi^{2}_{\rm max}$ distribution. The simplest and most effective amplitude
model used to fit the data, indicated as the nominal model in the following,
comprises a narrow ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}$
structure with spin-parity $J^{P}=1/2^{-}$, whose mass and width are extracted
from the amplitude fit, and two nonresonant (NR) contributions, one with
$J^{P}=1^{-}$ for the ${\Lambda}{\overline{\mathrm{p}}}$ system and a second
one with $J^{P}=1/2^{-}$ for the
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}}$. The
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}$ resonance is modelled
with a relativistic Breit–Wigner function as discussed in the Supplemental
material [SupplementalMaterial].

The couplings are defined in the $LS$ basis, both for the
${{{\mathrm{B}}^{-}}}\rightarrow XR$ process, and for the $R\rightarrow YZ$
process, where $X$, $Y$ and $Z$ are the final state particles, and
$R=K^{*-},{\overline{P}_{\psi}^{N}}^{-}$ and ${P_{\psi s}^{{\Lambda}}}^{0}$ is
the decay chain under consideration. Here, $L$ indicates the decay orbital
angular momentum and $S$ is the sum of the spins of the decay products. In the
nominal model $L=0$ is used for the production and decay of the narrow
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}$ resonance, while
$L=0,1,2$ and $L=0,2$ are used in the NR(${\Lambda}{\overline{\mathrm{p}}}$)
system for the production and decay, respectively, and $L=0$ and $L=1$ in the
NR(${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}}$)
system. Due to the small $Q$-value of the decay, higher values of the orbital
momentum are suppressed. Fixing the lowest orbital momentum couplings for the
NR(${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}}$) as
the normalization choice reduces the number of free parameters to sixteen: the
mass, the width and the complex coupling of the ${P_{\psi s}^{{\Lambda}}}^{0}$
resonant contribution; four complex couplings for the
NR(${\Lambda}{\overline{\mathrm{p}}}$) contribution; and a complex coupling
and two parameters for the second-order polynominal parameterization of the
lineshape for the
NR(${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}}$)
contribution.

A null-hypothesis model is used to test the significance of the ${P_{\psi
s}^{{\Lambda}}}^{0}$ state, which comprises only two NR contributions. nominal
model. The fit results for the nominal and the null-hypothesis model are shown
in Fig. 3. The null-hypothesis model does not describe the data, with a
corresponding $\chi^{2}_{\rm max}/{\rm n.d.f.}=120.8/47$. Using the nominal
model, a good fit to data was obtained with a $\chi^{2}_{\rm max}/{\rm
n.d.f.}=55.3/51$ and a $p$-value of $0.51$, computed by counting the number of
pseudoexperiments above the value of $\chi^{2}_{\rm max}$ observed in data.

Figure 3:  Distributions of invariant mass and $\cos\theta_{K^{*}}$ Fit
results to data using the nominal model are superimposed. The null-hypothesis
model fit results are also shown in grey. The $\varXi_{c}^{+}D^{-}$ baryon-
meson threshold at 4.337  GeV is indicated with a vertical dashed line in the
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda})$ invariant mass
distribution.

A new narrow ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}$
structure is observed with high significance in the nominal fit to data. Using
Wilks’ theorem, a statistical significance exceeding $15\sigma$ is estimated
from the value of $-2\Delta\log{\cal L}=243$ of the null-hypothesis model with
respect to the nominal model. The mass and width of the new pentaquark
candidate are measured to be $M_{P_{\psi s}^{{\Lambda}}}=4338.2\pm
0.7\text{\,Me\kern-1.00006ptV}$ and $\Gamma_{P_{\psi s}^{{\Lambda}}}=7.0\pm
1.2\text{\,Me\kern-1.00006ptV}$, respectively, where the uncertainties are
statistical only. This represents the first observation of a strange
pentaquark candidate with minimal quark content $\mathrm{c}$
$\overline{{\mathrm{c}}}$$\mathrm{u}$ $\mathrm{d}$ $\mathrm{s}$.

Alternative models are considered for systematic studies. To assess the
contribution of a ${\overline{P}_{\psi}^{N}}^{-}$ pentaquark candidate, a
relativistic Breit–Wigner function is used for the
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}})$
lineshape instead of a 2nd order polynomial function. The value of
$-2\Delta\log{\cal L}=80$ obtained with respect to the nominal fit indicates
that the
NR(${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}}$)
contribution is preferred over the hypothesis of a
${\overline{P}_{\psi}^{N}}^{-}$ candidate, while consistent results for the
$P_{\psi s}^{{\Lambda}}(4338)^{0}$ state parameters are obtained. The
contribution of a second narrow ${P_{\psi s}^{{\Lambda}}}^{0}$ resonance is
added to the nominal model to parametrize the
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda})$ distribution close
to the ${{\Lambda}^{+}_{\mathrm{c}}}{{\mathrm{D}}^{-}_{\mathrm{s}}}$ threshold
at $4255\text{\,Me\kern-1.00006ptV}$, and found to not be statistically
significant. Using the ${\rm CL}_{s}$ method [Read:2002hq], an upper limit on
the $P_{\psi s}^{{\Lambda}}(4255)^{0}$ fit fraction is set to 8.7% at a 95%
confidence level. To determine the $J^{P}$ assignments, all 16 combinations of
$J^{P}=1/2^{\pm},3/2^{\pm}$ are studied for the $P_{\psi
s}^{{\Lambda}}(4338)^{0}$ and
NR(${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}}$)
spin-parity hypotheses, and those with $-2\Delta\log{\cal L}>9$ with respect
to the nominal fit are discarded. For the $P_{\psi s}^{{\Lambda}}(4338)^{0}$
state, the $J^{P}=3/2^{\pm}$ hypotheses are discarded, the $J^{P}=1/2^{-}$
assignment is preferred, while the $J^{P}=1/2^{+}$ is excluded at a 90%
confidence level using the ${\rm CL}_{s}$ method [Read:2002hq].

Systematic uncertainties are evaluated on the mass and the width of the new
pentaquark candidate, and on the fit fractions of $P_{\psi
s}^{{\Lambda}}(4338)^{0}$,
NR(${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}}$) and
NR$({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda})$ contributions.
The uncertainties are summarised in Table 1 and are summed in quadrature for
the total contribution. For each systematic uncertainty, an ensemble of 1000
pseudoexperiments, generated according to the nominal model with the same
statistics as in data, is fitted with an alternative configuration that is
representative of the systematic effect. The uncertainty on each parameter is
determined as the mean value of the difference between the fit results of the
nominal and the alternative models. The main contributions are related to the
model for the decay amplitude, the bias of the fitting procedure, and the
uncertanty on the reconstruction efficiency
$\epsilon(m_{{\overline{\mathrm{p}}}{\Lambda}},\vec{\Omega})$. For the
amplitude model, the nominal value of the hadron radius for the
Blatt–Weisskopf coefficients [Blatt:1952ije] is assumed to be
$3\text{\,Ge\kern-1.00006ptV}^{-1}$ and varied to 1 and
$5\text{\,Ge\kern-1.00006ptV}^{-1}$, taking the largest effect as a systematic
uncertainty. Additional $LS$ couplings are considered with respect to the
nominal model, in particular the $L,S=1,1$ ($L,S=2,3/2$) coupling for the
production (decay) of $P_{\psi s}^{{\Lambda}}(4338)^{0}$ contribution, and the
$L,S=1,1$ coupling for the
NR$({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}})$
contribution. A relativistic Breit–Wigner function is used instead of the 2nd
order polynominal for the lineshape of the
NR$({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}})$
contribution. Moreover, a model with $J^{P}=1/2^{+}$ assignment to the
$P_{\psi s}^{{\Lambda}}(4338)^{0}$ state is also considered. Finally, the
behaviour of the maximum-likelihood estimator is studied using 1000
pseudoesperiments. Biases on the fit parameters are present due to the limited
sample size and are assigned as systematic uncertainties. For the
reconstruction efficiency, the nominal efficiency function based on decays
from either the long or downstream $\Lambda$ category and the largest effect
is considered as systematic uncertainty.

Additional systematic uncertainties account for the limited knowledge of the
${\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}$ decay amplitude parameters
[BESIII:2018cnd, PDG2022], the background parameterization, and the effect of
the resolution on the
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda})$ invariant mass.
The nominal background parameterization ${\cal P}_{\rm{bkg}}$ is obtained from
the distributions of candidates in the
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}})$
range ${[5200,5250]\cup[5340,5350]\text{\,Me\kern-1.00006ptV}}$, while the
parameterization obtained from the region
$[5295,5315]\text{\,Me\kern-1.00006ptV}$ is used to assess systematic effects.
The background fraction $\beta=0.07\pm 0.01$ is also varied within
uncertainties. The effect of the invariant mass resolution, about
$1\text{\,Me\kern-1.00006ptV}$ on average on
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda})$, is estimated by
smearing the invariant mass distributions of 1000 pseudoexperiments and
fitting them using the nominal model.

Table 1:  Systematic uncertainties on the mass ($M_{P_{\psi s}^{{\Lambda}}}$) and width ($\Gamma_{P_{\psi s}^{{\Lambda}}}$) of the ${P_{\psi s}^{{\Lambda}}}^{0}$ state (in  MeV), and on the fit fractions $f_{P_{\psi s}^{{\Lambda}}}$, $f_{\textrm{NR}({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}})}$ and $f_{\textrm{NR}({\Lambda}{\overline{\mathrm{p}}})}$ of the pentaquark candidate and nonresonant contributions (in %). Source | $M_{P_{\psi s}^{{\Lambda}}}$ | $\Gamma_{P_{\psi s}^{{\Lambda}}}$ | $f_{P_{\psi s}^{{\Lambda}}}$ | $f_{\textrm{NR}({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}})}$ | $f_{\textrm{NR}({\Lambda}{\overline{\mathrm{p}}})}$
---|---|---|---|---|---
Hadron radius | $0.1$ | $0.4$ | $0.3$ | $0.2$ | $0.2$
$LS$ values | $0.3$ | $0.1$ | $0.8$ | $0.7$ | $0.6$
Breit–Wigner ${\overline{P}_{\psi}^{N}}^{-}$ | $0.1$ | $0.9$ | $0.8$ | … | …
$J^{P}({P_{\psi s}^{{\Lambda}}}^{0})$ assignment | $0.1$ | $0.9$ | $1.2$ | $0.4$ | $0.9$
Fitting procedure | $0.1$ | $0.2$ | $0.1$ | $1.0$ | $1.1$
Efficiency | $0.02$ | $0.19$ | $0.02$ | $0.3$ | $0.2$
${\Lambda}$ decay parameters | $0.02$ | $0.04$ | $0.01$ | $0.3$ | $0.2$
Background | $0.01$ | $0.05$ | $0.96$ | $0.4$ | $0.7$
Mass resolution | $0.01$ | $0.03$ | $0.01$ | $0.1$ | $0.1$
Total | $0.4$ | $1.3$ | $1.9$ | $1.4$ | $1.7$

The mass and width of the new pentaquark candidate are measured to be
${M_{P_{\psi s}^{{\Lambda}}}=4338.2\pm 0.7\pm 0.4\text{\,Me\kern-1.00006ptV}}$
and $\Gamma_{P_{\psi s}^{{\Lambda}}}=7.0\pm 1.2\pm
1.3\text{\,Me\kern-1.00006ptV}$; the measured fit fractions are $f_{P_{\psi
s}^{{\Lambda}}}=0.125\pm 0.007\pm 0.019$, $f_{{\rm
NR}({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}})}=0.840\pm
0.022\pm 0.014$, and $f_{{\rm NR}({\Lambda}{\overline{\mathrm{p}}})}=0.113\pm
0.013\pm 0.017$, for the resonant ${P_{\psi s}^{{\Lambda}}}^{0}$ state, the
nonresonant
${\textrm{NR}({\mathrm{p}}{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}})}$,
and ${\textrm{NR}({\mathrm{p}}{\kern 0.0pt\overline{\kern 0.0pt\Lambda}})}$
contributions, respectively. The first uncertainty is statistical and the
second systematic. The $J^{P}=1/2^{-}$ quantum numbers for the ${P_{\psi
s}^{{\Lambda}}(4338)^{0}}$ state are preferred; $J=1/2$ is established and
positive parity can be excluded at 90% confidence level. Due to the small
$Q$-value of the decay, the most precise single measurement to date of the
${{\mathrm{B}}^{-}}$ mass, $5279.44\pm 0.05\pm
0.07\text{\,Me\kern-1.00006ptV}$, is performed. This measurement is based on
signal candidates with $\Lambda$ baryons in the long category. Systematic
uncertainties on the ${{\mathrm{B}}^{-}}$ mass include uncertainties on
particle interactions with the detector material
($0.030\text{\,Me\kern-1.00006ptV}$), momentum scaling due to imperfections in
the magnetic-field mapping ($0.039\text{\,Me\kern-1.00006ptV}$), and the
choice of the signal and background fit model
($0.050\text{\,Me\kern-1.00006ptV}$). Systematic uncertainties from knowledge
of the ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}},{\Lambda}$ and
${\mathrm{p}}$ masses are negligible.

In conclusion, an amplitude analysis of the
${{{\mathrm{B}}^{-}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}}$
decay is performed using about 4400 signal candidates, selected on data
collected by the LHCb experiment between 2011 and 2018 and corresponding to an
integrated luminosity of $9\text{\,fb}^{-1}$. A new resonant structure in the
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}$ system is found with
high statistical significance, representing the first observation of a
pentaquark candidate with strange quark content, named the ${P_{\psi
s}^{{\Lambda}}(4338)^{0}}$ state, with spin $J=1/2$ assigned and parity $P=-1$
preferred. No evidence for additional resonant states, either $P_{\psi
s}^{{\Lambda}}(4255)^{0}$ or ${\overline{P}_{\psi}^{N}}^{-}$ pentaquark
candidates or excited $K^{*-}$ resonances, is found from the fit to data. The
new ${P_{\psi s}^{{\Lambda}}(4338)^{0}}$ state is found at the threshold for
${{\Xi}^{+}_{\mathrm{c}}}{{\mathrm{D}}^{-}}$ baryon-meson production, which is
relevant for the interpretation of its nature.

## Supplemental Material

##  Appendix A Amplitude model

The amplitude model is constructed using helicity formalism [Chung:186421]
following the prescription for final particle spin matching described in Ref.
[DPdeco]. The amplitude $O^{X}_{\lambda_{1},\lambda_{2},\lambda_{3}}$
describes the decay amplitude for the ${{\mathrm{B}}^{-}}$ to the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ $\Lambda$
$\overline{\mathrm{p}}$ final state via the
$K^{*-},{\overline{P}_{\psi}^{N}}^{-},{P_{\psi s}^{{\Lambda}}}^{0}$ decay
chains as follows,

$\displaystyle
O^{K^{*}}_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}},\lambda_{{\Lambda}},\lambda_{\overline{\mathrm{p}}}}(m^{2}_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{{\Lambda}}},m^{2}_{{{\Lambda}}{\overline{\mathrm{p}}}})$
$\displaystyle=\sum_{j^{K^{*}}}\sum_{\\{\lambda^{\prime}\\}}\sqrt{\frac{2j^{K^{*}}+1}{4\pi}}\,\HellH{B^{-}\rightarrow
K^{*}{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}}_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime}}\,R(m^{2}_{{{\Lambda}}{\overline{\mathrm{p}}}})\,d^{j^{K^{*}}}_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime},\lambda_{{\Lambda}}^{\prime}-\lambda_{\overline{\mathrm{p}}}^{\prime}}(\theta_{K^{*}}$
$\displaystyle\times\HellH{K^{*}\rightarrow{{\Lambda}}{\overline{\mathrm{p}}}}_{\lambda_{{\Lambda}}^{\prime},\lambda_{\overline{\mathrm{p}}}^{\prime}}\delta_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime},\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}}\,d^{1/2}_{\lambda_{{\Lambda}}^{\prime},\lambda_{{\Lambda}}}(\wigner^{{{\Lambda}}}_{B{\overline{\mathrm{p}}}})\,d^{1/2}_{\lambda_{\overline{\mathrm{p}}}^{\prime},\lambda_{\overline{\mathrm{p}}}}(-\wigner^{{\overline{\mathrm{p}}}}_{B{{\Lambda}}})\times(-1)^{j^{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}}-\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime}}(-1)^{j^{\overline{\mathrm{p}}}-\lambda_{\overline{\mathrm{p}}}^{\prime}},$
$\displaystyle
O^{P_{\psi}^{N}}_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}},\lambda_{{\Lambda}},\lambda_{\overline{\mathrm{p}}}}(m^{2}_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{{\Lambda}}},m^{2}_{{{\Lambda}}{\overline{\mathrm{p}}}})$
$\displaystyle=\sum_{j^{P_{\psi}^{N}}}\sum_{\\{\lambda^{\prime}\\}}\sqrt{\frac{2j^{P_{\psi}^{N}}+1}{4\pi}}\,\HellH{B^{-}\rightarrow
P_{\psi}^{N}{{\Lambda}}}_{\lambda_{{\Lambda}}^{\prime}}\,R(m^{2}_{{\overline{\mathrm{p}}}{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}})\,d^{j^{P_{\psi}^{N}}}_{\lambda_{{\Lambda}}^{\prime},\lambda_{\overline{\mathrm{p}}}^{\prime}-\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime}}(\theta_{P_{\psi}^{N}})$
$\displaystyle\times\HellH{P_{\psi}^{N}\rightarrow{\overline{\mathrm{p}}}{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}}_{\lambda_{\overline{\mathrm{p}}}^{\prime},\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime}}d^{1}_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime},\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}}(-\wigner^{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}}_{B{\overline{\mathrm{p}}}})\,\delta_{\lambda_{{\Lambda}}^{\prime},\lambda_{{\Lambda}}}\,d^{1/2}_{\lambda_{\overline{\mathrm{p}}}^{\prime},\lambda_{\overline{\mathrm{p}}}}(\wigner^{{\overline{\mathrm{p}}}}_{B{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}})\times(-1)^{j^{{\Lambda}}-\lambda_{\Lambda}^{\prime}}(-1)^{j^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}-\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime}},$
$\displaystyle O^{{{P}_{\psi
s}^{{\Lambda}}}}_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}},\lambda_{{\Lambda}},\lambda_{\overline{\mathrm{p}}}}(m^{2}_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{{\Lambda}}},m^{2}_{{{\Lambda}}{\overline{\mathrm{p}}}})$
$\displaystyle=\sum_{j^{{P}_{\psi
s}^{{\Lambda}}}}\sum_{\\{\lambda^{\prime}\\}}\sqrt{\frac{2j^{{P}_{\psi
s}^{{\Lambda}}}+1}{4\pi}}\,\HellH{B^{-}\rightarrow{P}_{\psi
s}^{{\Lambda}}{\overline{\mathrm{p}}}}_{\lambda_{\overline{\mathrm{p}}}^{\prime}}\,R(m^{2}_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{{\Lambda}}})\,d^{J^{{P}_{\psi
s}^{{\Lambda}}}}_{\lambda_{\overline{\mathrm{p}}},\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime}-\lambda_{{\Lambda}}^{\prime}}(\theta_{{P}_{\psi
s}^{{\Lambda}}})$ $\displaystyle\times\HellH{{P}_{\psi
s}^{{\Lambda}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{{\Lambda}}}_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime},\lambda_{{\Lambda}}^{\prime}}d^{1}_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{\prime},\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}}(\wigner^{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}}_{B{{\Lambda}}})\,d^{1/2}_{\lambda_{{\Lambda}}^{\prime},\lambda_{{\Lambda}}}(-\wigner^{{{\Lambda}}}_{B{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}})\,\delta_{\lambda_{\overline{\mathrm{p}}}^{\prime},\lambda_{\overline{\mathrm{p}}}}\times(-1)^{j^{{\overline{\mathrm{p}}}}-\lambda_{\overline{\mathrm{p}}}^{\prime}}(-1)^{j^{{\Lambda}}-\lambda_{\Lambda}^{\prime}},$

where $j^{X}$ is the total angular momentum of the different contributions in
the ${X=K^{*-},\overline{P}_{\psi}^{N-}}$ and ${P_{\psi s}^{{\Lambda}}}^{0}$
decay chains, respectively, and $\\{\lambda^{\prime}\\}$ are the helicities of
the final particles before spin rotations. The angle, $\wigner^{i}_{Bk}$, is
between the $B^{-}$ and the particle $k$ in rest frame $i$. The coupling,
$H_{\lambda^{\prime}}^{A\rightarrow BC}$, is the helicity coupling of a two-
body decay $A\rightarrow BC$, $R$ is the line shape and
$d^{j}_{\lambda_{A},\lambda_{B}-\lambda_{C}}$ is the small Wigner function.
The angle, $\theta_{X}$, is the helicity angle of particle $X$, which is
calculated using the $\Lambda$ in the $K^{*-}$ rest frame, and either the
$\overline{\mathrm{p}}$ in the $\overline{P}_{\psi}^{N-}$ rest frame, or the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ in the $P_{\psi s}^{{\Lambda}}$
rest frame.

Figure 4:  Definition of the helicity angles used in the decay amplitudes. The
notation $\hat{p}^{X}_{Y}$ indicates the direction of the $X$ particle in the
$Y$ rest frame. The $\theta_{K^{*}}$ is the helicity angle of $\Lambda$ in the
rest frame of $K^{*}$, the $\theta_{P_{\psi}^{N}}$ is the helicity angle of
$\overline{\mathrm{p}}$ in the rest frame of $P_{\psi}^{N-}$, and the
$\theta_{P_{\psi s}^{{\Lambda}}}$ is the helicity angle of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$ in the rest frame of the
${P_{\psi s}^{{\Lambda}}}^{0}$.

The total decay amplitude is obtained by including the
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}\rightarrow{\upmu^{+}}\mu^{-}$
and the ${\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}$ decay amplitudes

$\displaystyle
A_{\lambda_{\overline{\mathrm{p}}},\lambda_{p},\Delta\mu}(m_{{\overline{\mathrm{p}}}{\Lambda}},\vec{\Omega})$
$\displaystyle=\sum_{\lambda_{\bar{\Lambda}},\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}}\left(O^{K^{*-}}+O^{P_{\psi}^{N-}}+O^{{P_{\psi
s}^{{\Lambda}}}^{0}}\right)_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}},\lambda_{{\Lambda}},\lambda_{\overline{\mathrm{p}}}}(m^{2}_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{{\Lambda}}},m^{2}_{{{\Lambda}}{\overline{\mathrm{p}}}})$
$\displaystyle\hskip 56.9055pt\times
D^{1*}_{\lambda_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}},\Delta_{\mu}}(\phi_{\mu^{-}},\theta_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}},0)\,\HellH{{{\Lambda}}\rightarrow{p}\pi}_{\lambda_{p}}D^{1/2*}_{\lambda_{{\Lambda}},\lambda_{p}}(\phi_{p},\theta_{{\Lambda}},0),$
(4)

where $D^{j*}_{\lambda_{A},\lambda_{B}-\lambda_{C}}(\phi,\theta,0)$ is the
Wigner $D$ matrix, equal to
$e^{i\lambda_{A}\phi}d^{j}_{\lambda_{A},\lambda_{B}-\lambda_{C}}(\theta)$,
$\phi_{\mu^{-}},\theta_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}},\phi_{p},\theta_{{\Lambda}}$
are the azimuthal and polar angles of $\mu^{-}$ and $\mathrm{p}$ in the
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}$ and ${\Lambda}$ rest frames,
respectively. The axes in the $\mathrm{B}$ rest frame are defined as follows,

$\displaystyle\begin{aligned}
\hat{x}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}&=\hat{y}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}\times\hat{z}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}},\\\
\hat{y}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}&=\frac{\vec{p}_{\overline{\mathrm{p}}}^{B}\times\vec{p}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{B}}{|\vec{p}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{B}\times\vec{p}_{\overline{\mathrm{p}}}^{B}|},\\\
\hat{z}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}&=\frac{\vec{p}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{B}}{|\vec{p}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}^{B}|},\\\
\end{aligned}$ $\displaystyle\begin{aligned}
\hat{x}_{{\Lambda}}&=\hat{y}_{{\Lambda}}\times\hat{z}_{{\Lambda}},\\\
\hat{y}_{{\Lambda}}&=\frac{\vec{p}_{\overline{\mathrm{p}}}^{B}\times\vec{p}_{{\Lambda}}^{B}}{|\vec{p}_{{\Lambda}}^{B}\times\vec{p}_{\overline{\mathrm{p}}}^{B}|},\\\
\hat{z}_{{\Lambda}}&=\frac{\vec{p}_{{\Lambda}}^{B}}{|\vec{p}_{{\Lambda}}^{B}|},\end{aligned}$
(5)

where the symbol $\hat{x}$ refers to $\vec{x}/|x|$. In Eq. 4, $\Delta\mu$ is
the difference of the muon helicities. For the
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}\rightarrow{\upmu^{+}}\mu^{-}$
decay, the coupling can be absorbed into the other couplings of the total
decay amplitude and therefore is not fit. Indeed, there is only one coupling
because the process with $\Delta\mu=0$ is highly suppressed. So, $\Delta\mu$
can only take values $1$ and $-1$, and both choices lead to the same helicity
coupling due to parity conservation.

Enforcing $C\\!P$ conservation, the helicity couplings for $B^{-}$ and
${{\mathrm{B}}^{+}}$ decays are the same. The matrix-element formula is the
same for charge-conjugate decays, but all azimuthal angles must change sign
due to charge-parity transformation, i.e.
$\phi_{p}\rightarrow-\phi_{{\overline{\mathrm{p}}}}$ and
$\phi_{\mu^{-}}\rightarrow-\phi_{\mu^{+}}$.

The ${\kern 0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}$ decay
parameters are defined by

$\displaystyle\alpha_{+}=\frac{|H^{{\kern 0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{1/2}|^{2}-|H^{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{-1/2}|^{2}}{|H^{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{1/2}|^{2}+|H^{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{-1/2}|^{2}},$
$\displaystyle\beta_{+}=\frac{2\textrm{Im}(H^{{\kern 0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{1/2}H^{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}*}_{-1/2})}{|H^{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{1/2}|^{2}+|H^{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{-1/2}|^{2}},$
(6) $\displaystyle\gamma_{+}=\frac{2\textrm{Re}(H^{{\kern 0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{1/2}H^{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}*}_{-1/2})}{|H^{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{1/2}|^{2}+|H^{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{-1/2}|^{2}},$

which satisfy the relation,

$\alpha^{2}_{+}+\beta^{2}_{+}+\gamma^{2}_{+}=1.$

It is convenient to express $\beta_{+}$ and $\gamma_{+}$ in terms of an angle
$\phi_{+}$ defined as

$\displaystyle\beta_{+}=\sqrt{1-\alpha^{2}_{+}}\sin{\phi_{+}},$ (7)
$\displaystyle\gamma_{+}=\sqrt{1-\alpha^{2}_{+}}\cos{\phi_{+}}.$

Enforcing $C\\!P$ conservation, the following relations hold,

$\displaystyle H^{{\kern 0.0pt\overline{\kern
0.0pt\Lambda}}\rightarrow{\overline{\mathrm{p}}}{{\uppi}^{+}}}_{\mp
1/2}=\eta_{{\Lambda}}\eta_{{\mathrm{p}}}\eta_{\pi}(-1)^{J_{{\Lambda}}-J_{{\mathrm{p}}}-J_{\pi}}H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{\pm
1/2}=-H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{\pm 1/2}.$ (8)

This leads to

$\displaystyle\alpha_{+}=-\alpha_{-},$ $\displaystyle\beta_{+}=-\beta_{-},$
$\displaystyle\gamma_{+}=\gamma_{-},$
$\displaystyle\tan{\phi_{+}}=-\tan{\phi_{-}},$ (9)

where $\alpha_{-}$, $\beta_{-}$ and $\gamma_{-}$ are obtained following Eq. A
but using the couplings of the conjugate decay.

$\displaystyle\alpha_{-}=\frac{|H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{1/2}|^{2}-|H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{-1/2}|^{2}}{|H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{1/2}|^{2}+|H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{-1/2}|^{2}},$
$\displaystyle\beta_{-}=\frac{2\textrm{Im}(H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{1/2}H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}*}_{-1/2})}{|H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{1/2}|^{2}+|H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{-1/2}|^{2}},$
$\displaystyle\gamma_{-}=\frac{2\textrm{Re}(H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{1/2}H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}*}_{-1/2})}{|H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{1/2}|^{2}+|H^{{\Lambda}\rightarrow{\mathrm{p}}{{\uppi}^{-}}}_{-1/2}|^{2}}.$

If the complex helicity coupling $H^{\LbarTopbarpip}_{1/2}$ is set to (1,0),
then
$H^{\LbarTopbarpip}_{-1/2}=\sqrt{\frac{1-\alpha_{+}}{1+\alpha_{+}}}e^{-i\phi_{+}}$.
The values of $\alpha_{+}$ and $\phi_{+}$ are fixed to $-0.758$
[BESIII:2018cnd] and $+6.5^{o}$ [PDG2022] respectively.

The helicity couplings for the decay $A\rightarrow BC$ can be expressed as a
combination of the $LS$ couplings ($B_{L,S}$) using the Clebsch–Gordan (CG)
coefficients

$\displaystyle H^{A\rightarrow
BC}_{\lambda_{B},\lambda_{C}}=\sum_{L}\sum_{S}\sqrt{\frac{2L+1}{2J_{A}+1}}B_{L,S}$
$\displaystyle\langle{J_{B}},{\lambda_{B}},{J_{C}},{-\lambda_{C}}|{S},{\lambda_{B}-\lambda_{C}}\rangle$
$\displaystyle\times$
$\displaystyle\langle{L},{0},{S},{\lambda_{B}-\lambda_{C}}|{J_{A}},{\lambda_{B}-\lambda_{C}}\rangle,$
(10)

where $L$ is the orbital angular momentum in the decay, and $S$ is the total
spin of the daughters, $\vec{S}=\vec{J}_{B}+\vec{J}_{C}$ ($|J_{B}-J_{C}|\leq
S\leq J_{B}+J_{C}$). If the $Q$ value, defined as $Q=M_{A}-M_{B}-M_{C}$, is
small ($Q/M_{A}\ll 1$), the higher orbital angular momenta are suppressed,
hence the number of couplings is reduced. CG coefficients automatically take
into account parity conservation constraints on helicity couplings for a
strong or electromagnetic decay.

For ${{{\mathrm{B}}^{-}}}\rightarrow XR$, $R\rightarrow YZ$ cascade decay,
e.g. $X={\overline{\mathrm{p}}}$, $R={P_{\psi s}^{{\Lambda}}}^{0}$,
$Y={\Lambda}$ and $Z={{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}$, the
lineshape of $R$ is

$\displaystyle\left(\frac{p}{p_{0}}\right)^{L}B^{\prime}_{L}(p,p_{0},d)\times\left(\frac{q}{q_{0}}\right)^{l}B^{\prime}_{l}(q,q_{0},d)\textrm{BW}(m|m_{0},\Gamma_{0}),$
(11)

where $p$ is the momentum of resonance $R$ in the ${{\mathrm{B}}^{-}}$ rest
frame, $q$ is the momentum of particle $Y$ in the rest frame of resonance $R$,
$p_{0}$ and $q_{0}$ are the momentum values calculated at the $R$ resonance
peak, $L$ is the orbital angular momentum between resonance $R$ and particle
$X$ in the ${{{\mathrm{B}}^{-}}}\rightarrow XR$ decay, and $l$ is the orbital
angular momentum between particle $Y$ and particle $Z$ in the $R\rightarrow
YZ$ decay. The $\left(p/p_{0}\right)^{L}$ and $\left(q/q_{0}\right)^{l}$
contributions are the orbital barrier factors, $B^{\prime}_{L}(p,p_{0},d)$ and
$B^{\prime}_{l}(q,q_{0},d)$ are the Blatt–Weisskopf functions that account for
the difficulty to create the orbital angular momentum, and depend on the
production (decay) momentum $p$ ($q$) and on the size of the decaying particle
given by the hadron radius $d$. These coefficients up to order 4 are listed
below,

$\displaystyle B_{0}^{\prime}\left(p,p_{0},d\right)=1,$ $\displaystyle
B_{1}^{\prime}\left(p,p_{0},d\right)=\sqrt{\frac{1+\left(p_{0}d\right)^{2}}{1+(pd)^{2}}},$
$\displaystyle
B_{2}^{\prime}\left(p,p_{0},d\right)=\sqrt{\frac{9+3\left(p_{0}d\right)^{2}+\left(p_{0}d\right)^{4}}{9+3(pd)^{2}+(pd)^{4}}},$
$\displaystyle
B_{3}^{\prime}\left(p,p_{0},d\right)=\sqrt{\frac{225+45\left(p_{0}d\right)^{2}+6\left(p_{0}d\right)^{4}+\left(p_{0}d\right)^{6}}{225+45(pd)^{2}+6(pd)^{4}+(pd)^{6}}},$
$\displaystyle
B_{4}^{\prime}\left(p,p_{0},d\right)=\sqrt{\frac{11025+1575\left(p_{0}d\right)^{2}+135\left(p_{0}d\right)^{4}+10\left(p_{0}d\right)^{6}+\left(p_{0}d\right)^{8}}{11025+1575(pd)^{2}+135(pd)^{4}+10(pd)^{6}+(pd)^{8}}},$
(12)

where $d$ is the particle size parameter, set to
$3\text{\,Ge\kern-1.00006ptV}^{-1}$ following the convention of Ref. [LHCb-
PAPER-2015-029]. In the nominal amplitude fit of
${{{\mathrm{B}}^{-}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}}$
decays, the constant $d$ is set to $d_{B}=d_{R}=3$ GeV -1 for the
${{\mathrm{B}}^{-}}$ and intermediate resonant $R$ decays.

The relativistic Breit–Wigner amplitude is given by

$\displaystyle\textrm{BW}(m|m_{0},\Gamma_{0})=\frac{1}{m^{2}_{0}-m^{2}-im_{0}\Gamma(m)},$
(13)

with

$\displaystyle\Gamma(m)=\Gamma_{0}\left(\frac{q}{q_{0}}\right)^{2l+1}\left(\frac{m_{0}}{m}\right)B^{\prime}_{l}\left(q,q_{0},d\right)^{2},$
(14)

where $m$ is the invariant mass of the $YZ$ system, and $m_{0}$ ($\Gamma_{0}$)
is the mass (width) of the $R$ resonance. In the case that resonance $R$ has a
mass peak outside of the accessible kinematic region, i.e.
$m_{R}>m_{{{\mathrm{B}}^{-}}}-m_{X}$, such as for the $K^{*}_{2}(2250)^{-}$
and $K^{*}_{3}(2320)^{-}$ states, the effective mass $m^{\textrm{eff}}_{0}$ is
introduced to calculate the two-body-decay momentum $q_{0}$ in Eq. 14,

$\displaystyle
m^{\textrm{eff}}_{0}(m_{0})=m^{\textrm{min}}+\frac{1}{2}(m^{\textrm{max}}-m^{\textrm{min}})\left[1+\tanh\left(\frac{m_{0}-\frac{m^{\textrm{min}}+m^{\textrm{max}}}{2}}{m^{\textrm{max}}-m^{\textrm{min}}}\right)\right].$
(15)

This term is a constant that can be absorbed into the couplings, since it
enters only in Eq. 14, and the mass $m_{0}$ and width $\Gamma_{0}$ of the
$K^{*}$ resonant contributions are fixed to the nominal values [PDG2022]. In
the case of a resonance $R$ with mass peak located outside of the phase space
at values $m_{R}<m_{Y}+m_{Z}$, such as for the $K^{*}_{4}(2045)^{-}$ state,
the width is chosen as mass-independent parameter $\Gamma_{0}$. In the nominal
model, the non-resonant (NR) contribution is modelled by a second-order
polynomial,

$c_{0}+c_{1}(m-m_{0})+c_{2}(m-m_{0})^{2},$ (16)

where $m_{0}$ is the average value of the invariant mass distribution, i.e. of
the $m_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}}}$
invariant mass distribution. The coefficients, $c_{i}$, are the polynomial
coefficients, where $c_{0}$ is set to a constant value since one of the
$c_{i}$ coefficients can be factor out of amplitude matrix element, and the
other two are extracted from a fit to the data.

##  Appendix B Event-by-event efficiency parameterisation

Event-by-event acceptance corrections are applied to the data using an
efficiency parameterisation based on the decay kinematics. The 6-body phase
space of the topology
${{{\mathrm{B}}^{-}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}(\rightarrow{\upmu^{-}}{\upmu^{+}}){\Lambda}(\rightarrow{\mathrm{p}}{{\uppi}^{-}}){\overline{\mathrm{p}}}$
is fully described by six independent kinematic variables:
$m_{{\Lambda}{\overline{\mathrm{p}}}}$, $\cos\theta_{K^{*}}$,
$\cos\theta_{J/\psi}$, $\phi_{\mu}$, $\cos\theta_{{\kern 0.0pt\overline{\kern
0.0pt\Lambda}}}$, and $\phi_{{\overline{\mathrm{p}}}}$. For the signal mode,
the overall efficiency, including trigger, detector acceptance, and selection
procedure, is obtained from simulation as a function of the six kinematic
variables,
$\vec{\omega}\equiv\\{\cos\theta_{K^{*}},\cos\theta_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}},\phi^{\prime}_{\mu},m^{\prime}_{{\Lambda}{\overline{\mathrm{p}}}},\cos\theta_{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}},\phi^{\prime}_{{\overline{\mathrm{p}}}}\\}$. Here,
$m^{\prime}_{{\Lambda}{\overline{\mathrm{p}}}}$ and $\phi^{\prime}$ are
transformed such that all four variables in $\vec{\omega}$ lie in the range
$(-1,1]$. The efficiency is parameterised as the product of Legendre
polynomials

$\displaystyle\epsilon(\vec{\omega})=\sum_{i,j,k,l,m,n}$ $\displaystyle
c_{i,j,k,l,m,n}P(\cos\theta_{K^{*}},i)P(\cos\theta_{J/\psi},j)$ (17)
$\displaystyle
P(\phi^{\prime}_{\mu},k)P(m^{\prime}_{{\Lambda}{\overline{\mathrm{p}}}},l)P(\cos\theta_{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}},m)P(\phi^{\prime}_{{\overline{\mathrm{p}}}},n),$

where $P(x,l)$ are Legendre polynomials of order $l$ in $x\in(-1,1]$.
Employing the order of the polynomials as $\\{2,2,2,2,4,3\\}$ for
$\\{\cos\theta_{K^{*}},\cos\theta_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}},\phi^{\prime}_{\mu},m^{\prime}_{{\Lambda}{\overline{\mathrm{p}}}},\cos\theta_{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}},\phi^{\prime}_{{\overline{\mathrm{p}}}}\\}$, respectively, was
found to give a good parameterisation. The coefficients $c_{i,j,k,l,m,n}$ are
determined from a moment analysis of
${{{\mathrm{B}}^{-}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}}$
phase-space simulated samples

$\displaystyle
c_{i,j,k,l,m,n}=\frac{C}{\sum_{\nu}\omega_{\nu}}\sum^{N_{rec}}_{\nu=1}\omega_{\nu}\left(\frac{2i+1}{2}\right)\left(\frac{2j+1}{2}\right)\left(\frac{2k+1}{2}\right)\left(\frac{2l+1}{2}\right)\left(\frac{2m+1}{2}\right)\left(\frac{2n+1}{2}\right)$
(18) $\displaystyle\times
P(\cos\theta_{K^{*}},i)P(\cos\theta_{J/\psi},j)P(\phi^{\prime}_{\mu},k)P(m^{\prime}_{{\Lambda}{\overline{\mathrm{p}}}},l)P(\cos\theta_{{\kern
0.0pt\overline{\kern
0.0pt\Lambda}}},m)P(\phi^{\prime}_{{\overline{\mathrm{p}}}},n),$

where $\omega_{\nu}$ is the per-event weight taking into account both the
generator-level phase-space element, $\mathrm{d}\Phi$, and the kinematic event
weights. Simulation samples are employed where
${{{\mathrm{B}}^{-}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda}{\overline{\mathrm{p}}}$
events are generated uniformly in phase space. In order to render the
simulation flat also in $m({\Lambda}{\overline{\mathrm{p}}})$, the inverted
phase-space factor, $1/\mathrm{d}\Phi$, is considered. The factors of
$(2a+1)/2$ arise from the orthogonality of the Legendre polynomials,

$\int_{-1}^{+1}P(x,a)P(x,a^{\prime})\mathrm{d}x=\frac{2}{2a+1}\delta_{aa^{\prime}}~{}.$
(19)

The sum in Eq. 18 is over the reconstructed events in the simulation sample
after all selection criteria. The factor $C$ ensures appropriate normalisation
and it is computed such that

$\displaystyle\sum_{n=0}^{N_{\rm gen}}\varepsilon(\vec{x}_{n})=N_{\rm rec},$
(20)

where $N_{\rm rec}$ is the total number of reconstructed signal events.

Up to statistical fluctuations, the parameterisation follows the simulated
data in all the distributions.

##  Appendix C Fit results of the nominal model

In Table 2, the fit results of the nominal model are reported including the
results of the $LS$ couplings. The couplings are split into real and imaginary
parts, i.e. Re${}_{\textrm{prod(decay)}}(R)_{L,S}$,
Im${}_{\textrm{prod(decay)}}(R)_{L,S}$. The subscript prod (decay) refers to
the ${{{\mathrm{B}}^{-}}}\rightarrow XR$ ($R\rightarrow YZ$) process, where
$X$, $Y$, $Z$ are the final state particles, and $R$ is the decay chain under
consideration. The subscript $L$ refers to the orbital angular momentum and
$S$ to the sum of the spins of the decay products.

Table 2: Parameters determined from the fit to data using the nominal model where uncertainties are statistical only. Parameters | Values
---|---
$M_{P_{\psi s}^{{\Lambda}}}$ ( MeV) | $4338.2\pm 0.7$
$\Gamma_{P_{\psi s}^{{\Lambda}}}$ ( MeV) | $7.0\pm 1.0$
Re${}_{\textrm{decay}}$(${P}_{\psi s}^{{\Lambda}\ 0}$)L=0,S=1/2 | $0.16\pm 0.04$
Im${}_{\textrm{decay}}$(${P}_{\psi s}^{{\Lambda}\ 0}$)L=0,S=1/2 | $-0.04\pm 0.08$
Re${}_{\textrm{decay}}$(NR($\overline{\mathrm{p}}$ ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$))L=1,S=3/2 | $-3.0\pm 0.4$
Im${}_{\textrm{decay}}$(NR($\overline{\mathrm{p}}$ ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}$))L=1,S=3/2 | $0.1\pm 0.5$
Re${}_{\textrm{prod}}$(NR($\overline{\mathrm{p}}$ $\Lambda$))L=1,S=1 | $1.0\pm 0.5$
Im${}_{\textrm{prod}}$(NR($\overline{\mathrm{p}}$ $\Lambda$))L=1,S=1 | $-0.5\pm 0.5$
Re${}_{\textrm{prod}}$(NR($\overline{\mathrm{p}}$ $\Lambda$))L=2,S=2 | $0.01\pm 0.25$
Im${}_{\textrm{prod}}$(NR($\overline{\mathrm{p}}$ $\Lambda$))L=2,S=2 | $0.1\pm 0.4$
Re${}_{\textrm{decay}}$(NR($\overline{\mathrm{p}}$ $\Lambda$))L=0,S=1 | $-0.3\pm 0.1$
Im${}_{\textrm{decay}}$(NR($\overline{\mathrm{p}}$ $\Lambda$))L=0,S=1 | $-0.1\pm 0.1$
Re${}_{\textrm{decay}}$(NR($\overline{\mathrm{p}}$ $\Lambda$))L=2,S=1 | $0.4\pm 0.1$
Im${}_{\textrm{decay}}$(NR($\overline{\mathrm{p}}$ $\Lambda$))L=2,S=1 | $0.1\pm 0.2$
$c_{1}$ | $2.6\pm 0.6$
$c_{2}$ | $72\pm 14$
$f_{\textrm{$P_{\psi s}^{{\Lambda}}$}}$ | $0.125\pm 0.007$
$f_{\textrm{NR}({\overline{\mathrm{p}}}{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}})}$ | $0.840\pm 0.002$
$f_{\textrm{NR}({\overline{\mathrm{p}}}{\Lambda})}$ | $0.113\pm 0.013$
$-\log{L}$ | $-807.63$

##  Appendix D Angular moments

The normalized angular moments $\left<P_{j}^{U}\right>$ of the ${{P}_{\psi
s}^{{\Lambda}}}^{0}$ helicity angle are defined as,

$\left<P_{j}^{U}\right>=\sum_{i=0}^{N_{rec}}\omega_{i}P_{j}\left(\cos\theta_{{P}_{\psi
s}^{{\Lambda}}}\right)$ (42)

where $N_{rec}$ is the number of selected events, $P_{j}$ are Legendre
polynomials and $\omega_{i}$ are per-event weights accounting for background
subtraction (with sPlot technique) and efficiency correction.

The angular moments are shown in Fig. D, up to order 5, as a function of the
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda})$ invariant mass
distribution. They show a good agreement between the data and the nominal
model.

Figure 5: ${P}_{\psi s}^{{\Lambda}}(4338)^{0}$ helicity angular moments as a
function of $m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda})$
invariant mass. The black points represent the data while the blue line is the
nominal model.

##  Appendix E Efficiency corrected and background subtracted distributions

The data are assigned weights to account for the efficiency and to subtract
the background using the sPlot technique. The efficiency corrected data
distributions of $m({\overline{\mathrm{p}}}{\Lambda})$,
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\overline{\mathrm{p}}})$,
$m({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{\Lambda})$ and
$\cos{\theta_{K^{*}}}$ are shown in Figure E. There is a sign difference
between this $\cos{\theta_{K^{*}}}$ definition and the one from CMS
[CMS:2019kbn].

Figure 6: Efficiency corrected and background subtracted distributions for
$m({\overline{\mathrm{p}}}\Lambda)$, $m(J/\psi{\overline{\mathrm{p}}})$,
$m(J/\psi\Lambda)$ and $\cos{\theta_{K^{*}}}$.

The author list for journal publications is generated from the Membership
Database shortly after ’approval to go to paper’ has been given. It will be
sent to you by email shortly after a paper number has been assigned. The
author list should be included in the draft used for first and second
circulation, to allow new members of the collaboration to verify that they
have been included correctly. Occasionally a misspelled name is corrected, or
associated institutions become full members. Therefore an updated author list
will be sent to you after the final EB review of the paper. In case line
numbering doesn’t work well after including the authorlist, try moving the
`\bigskip` after the last author to a separate line.

The authorship for Conference Reports should be “The LHCb collaboration”, with
a footnote giving the name(s) of the contact author(s), but without the full
list of collaboration names.

The authorship for Figure Reports should be “The LHCb collaboration”, with no
contact author and without the full list of collaboration names.