EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-156 LHCb-PAPER-2016-019 29 September 2016

Amplitude analysis of $B^{+}\rightarrow J/\psi\phi K^{+}$ decays

The LHCb collaboration†††Authors are listed at the end of this paper.

The first full amplitude analysis of $B^{+}\rightarrow J/\psi\phi K^{+}$ with
$J/\psi\rightarrow\mu^{+}\mu^{-}$, $\phi\rightarrow K^{+}K^{-}$ decays is
performed with a data sample of 3 fb-1 of $pp$ collision data collected at
$\sqrt{s}=7$ and $8$ TeV with the LHCb detector. The data cannot be described
by a model that contains only excited kaon states decaying into $\phi K^{+}$,
and four $J/\psi\phi$ structures are observed, each with significance over $5$
standard deviations. The quantum numbers of these structures are determined
with significance of at least $4$ standard deviations. The lightest has mass
consistent with, but width much larger than, previous measurements of the
claimed $X(4140)$ state. The model includes significant contributions from a
number of expected kaon excitations, including the first observation of the
$K^{*}(1680)^{+}\rightarrow\phi K^{+}$ transition.

Published in Physical Review D 95, 012002 (2017).

© CERN on behalf of the LHCb collaboration, license CC-BY-4.0.

##  1 Introduction

In 2008 the CDF collaboration presented $3.8\sigma$ evidence for a near-
threshold $X(4140)\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ mass peak in
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decays111Inclusion of charge-conjugate processes is implied throughout this
paper, unless stated otherwise. also referred to as $Y(4140)$ in the
literature, with width $\Gamma=11.7\mathrm{\,Me\kern-1.00006ptV}$ [1].222Units
with $c=1$ are used. Much larger widths are expected for charmonium states at
this mass because of open flavor decay channels [2], which should also make
the kinematically suppressed
$X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ decays
undetectable. Therefore, the observation by CDF triggered wide interest. It
has been suggested that the $X(4140)$ structure could be a molecular state [3,
4, 5, 6, 7, 8, 9, 10, 11], a tetraquark state [12, 13, 14, 15, 16], a hybrid
state [17, 18] or a rescattering effect [19, 20].

The LHCb collaboration did not see evidence for the narrow $X(4140)$ peak in
the analysis presented in Ref. [21], based on a data sample corresponding to
0.37 fb-1 of integrated luminosity, a fraction of that now available. Searches
for the narrow $X(4140)$ did not confirm its presence in analyses performed by
the Belle [22, 23] (unpublished) and BaBar [24] experiments. The $X(4140)$
structure was observed however by the CMS ($5\sigma$) [25] collaboration.
Evidence for it was also reported in
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decays by the D0 ($3\sigma$) [26] collaboration. The D0 collaboration claimed
in addition a significant signal for prompt $X(4140)$ production in $p\bar{p}$
collisions [27]. The BES-III collaboration did not find evidence for
$X(4140)\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ in
$e^{+}e^{-}\rightarrow\gamma X(4140)$ and set upper limits on its production
cross-section at $\sqrt{s}=4.23$, $4.26$ and $4.36$
$\mathrm{\,Ge\kern-1.00006ptV}$ [28]. Previous results related to the
$X(4140)$ structure are summarized in Table 1.

Table 1: Previous results related to the
$X(4140)\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ mass
peak, first observed in
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decays. The first (second) significance quoted for Ref. [27] is for the prompt
(non-prompt) production components. The statistical and systematic errors are
added in quadrature and then used in the weights to calculate the averages,
excluding unpublished results (shown in italics). The last column gives a
fraction of the total $B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi K^{+}$ rate attributed to the $X(4140)$ structure.

Year Experiment $B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi K$ $X(4140)$ peak luminosity yield Mass
[$\mathrm{\,Me\kern-0.80005ptV}$ ] Width [$\mathrm{\,Me\kern-0.80005ptV}$ ]
Sign. Fraction % 2008 CDF 2.7 fb-1 [1] $58\pm 10$
$4143.0\\!\pm\\!2.9\\!\pm\\!1.2$  $11.7\,^{+8.3}_{-5.0}\\!\pm\\!3.7$
$3.8\sigma$ 2009 Belle [22] $\mathit{325\pm 21}$ $\mathit{4143.0}$ fixed
$\mathit{11.7}$ fixed $\mathit{1.9\sigma}$ 2011 CDF 6.0 fb-1 [29]
$\mathit{115\pm 12}$ $\mathit{4143.4\,^{+2.9}_{-3.0}\\!\pm\\!0.6}$
$\mathit{15.3\\!^{+10.4}_{-\phantom{0}6.1}\\!\\!\pm\\!\\!2.5}$
$\mathit{5.0\sigma}$ $\mathit{14.9\\!\pm\\!3.9\\!\pm\\!2.4}$ 2011 LHCb 0.37
fb-1 [21] $346\pm 20$ $4143.4$ fixed $15.3$ fixed $1.4\sigma$ $<7$ @ 90%CL
2013 CMS 5.2 fb-1 [25] $2480\pm 160$ $4148.0\\!\pm\\!2.4\\!\pm\\!6.3$
$28\phantom{.0}\,^{+15\phantom{.0}}_{-11\phantom{.0}}\pm 19$ $5.0\sigma$
$10\\!\pm\\!3$ (stat.) 2013 D0 10.4 fb-1 [26] $215\pm 37$
$4159.0\\!\pm\\!4.3\\!\pm\\!6.6$ $19.9\\!\pm\\!12.6\,^{+1.0}_{-8.0}$
$3.0\sigma$ $21\\!\pm\\!8\\!\pm\\!4$ 2014 BaBar [24] $189\pm 14$ $4143.4$
fixed $15.3$ fixed $1.6\sigma$ $<13.3$ @ 90%CL 2015 D0 10.4 fb-1 [27]
$p\bar{p}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi...$ $4152.5\\!\pm\\!1.7\,^{+6.2}_{-5.4}$
$16.3\\!\pm\\!5.6\\!\pm\\!11.4$ $4.7\sigma$ ($5.7\sigma$)  Average
$4147.1\\!\pm\\!2.4$ $15.7\\!\pm\\!6.3$

In an unpublished update to their
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
analysis [29], the CDF collaboration presented $3.1\sigma$ evidence for a
second relatively narrow ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
mass peak near $4274\mathrm{\,Me\kern-1.00006ptV}$. This observation has also
received attention in the literature [30, 31]. A second
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ mass peak was observed by
the CMS collaboration at a mass which is higher by $3.2$ standard deviations,
but the statistical significance of this structure was not determined [25].
The Belle collaboration saw $3.2\sigma$ evidence for a narrow
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ peak at
$4350.6\,^{+4.6}_{-5.1}\pm 0.7\mathrm{\,Me\kern-1.00006ptV}$ in two-photon
collisions, which implies $J^{PC}=0^{++}$ or $2^{++}$, and found no evidence
for $X(4140)$ in the same analysis [32]. The experimental results related to
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ mass peaks heavier than
$X(4140)$ are summarized in Table 2.

Table 2: Previous results related to ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ mass structures heavier than the $X(4140)$ peak. The unpublished
results are shown in italics.

Year Experiment $B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi K$ $X(4274-4351$) peaks(s) luminosity yield Mass
[$\mathrm{\,Me\kern-0.80005ptV}$ ] Width [$\mathrm{\,Me\kern-0.80005ptV}$ ]
Sign. Fraction [%] 2011 CDF 6.0 fb-1 [29] $\mathit{115\pm 12}$
$\mathit{4274.4\,^{+8.4}_{-6.7}\\!\pm\\!1.9}$
$\mathit{32.3{{}^{+21.9}_{-15.3}}\\!\\!\pm\\!7.6}$ $\mathit{3.1\sigma}$ 2011
LHCb 0.37 fb-1 [21] $346\pm 20$ $4274.4$ fixed $32.3$ fixed $<\phantom{0}8$ @
90%CL 2013 CMS 5.2 fb-1 [25] $2480\pm 160$ $4313.8\\!\pm\\!5.3\\!\pm\\!7.3$
 $38\phantom{.0}\,^{+30\phantom{.0}}_{-15\phantom{.0}}\\!\pm\\!16$ 2013 D0
10.4 fb-1 [26] $215\pm 37$ $4328.5\\!\pm\\!12.0$ $30$ fixed 2014 BaBar [24]
$189\pm 14$ $4274.4$ fixed $32.3$ fixed $1.2\sigma$ $<18.1$ @ 90%CL 2010 Belle
[32] $\gamma\gamma\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ $4350.6\,^{+4.6}_{-5.1}\\!\pm\\!0.7$
$13\,^{+18}_{-\phantom{0}9}\\!\pm\\!4$ $3.2\sigma$

In view of the considerable theoretical interest in possible exotic hadronic
states decaying to ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$, it is
important to clarify the rather confusing experimental situation concerning
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ mass structures. The data
sample used in this work corresponds to an integrated luminosity of $3$ fb-1
collected with the LHCb detector in $pp$ collisions at center-of-mass energies
7 and 8 $\mathrm{\,Te\kern-1.00006ptV}$. Thanks to the larger signal yield,
corresponding to $4289\pm 151$ reconstructed
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decays, the roughly uniform efficiency and the relatively low background
across the entire ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ mass
range, this data sample offers the best sensitivity to date, not only to probe
for the $X(4140)$, $X(4274)$ and other previously claimed structures, but also
to inspect the high mass region.

All previous analyses were based on naive
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ mass
($m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$) fits, with
Breit–Wigner signal peaks on top of incoherent background described by ad-hoc
functional shapes (e.g. three-body phase space distribution in
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decays). While the $m_{\phi K}$ distribution has been observed to be smooth,
several resonant contributions from kaon excitations (hereafter denoted
generically as $K^{*}$) are expected. It is important to prove that any
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$ peaks are not merely
reflections of these conventional resonances. If genuine
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ states are present, it is
crucial to determine their quantum numbers to aid their theoretical
interpretation. Both of these tasks call for a proper amplitude analysis of
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decays, in which the observed $m_{\phi K}$ and
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$ masses are analyzed
simultaneously with the distributions of decay angles, without which the
resolution of different resonant contributions is difficult, if not
impossible. The analysis of ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
and $\phi$ polarizations via their decays to $\mu^{+}\mu^{-}$ and
$K^{+}K^{-}$, respectively, increases greatly the sensitivity of the analysis
as compared with the Dalitz plot analysis alone. In addition to the search for
exotic hadrons, which includes
$X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ and
$Z^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$
contributions, the amplitude analysis of our data offers unique insight into
the spectroscopy of the poorly experimentally understood higher excitations of
the kaon system, in their decays to a $\phi K^{+}$ final state.

In this article, an amplitude analysis of the decay
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$ is
presented for the first time, with additional results for, and containing more
detailed description of, the work presented in Ref. [33].

##  2 LHCb detector

The LHCb detector [34, 35] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of momentum, $p$, of
charged particles with relative uncertainty that varies from 0.5% at low
momentum to 1.0% at $200\mathrm{\,Ge\kern-1.00006ptV}$. The minimum distance
of a track to a primary vertex (PV), the impact parameter (IP), is measured
with a resolution of $(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$,
where $p_{\mathrm{T}}$ is the component of the momentum transverse to the
beam, in $\mathrm{\,Ge\kern-1.00006ptV}$. Different types of charged hadrons
are distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The online event selection is performed by a trigger, which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction.

##  3 Data selection

Candidate events for this analysis are first required to pass the hardware
trigger, which selects muons with transverse momentum
$\mbox{$p_{\mathrm{T}}$}>1.48\mathrm{\,Ge\kern-1.00006ptV}$ in the
7$\mathrm{\,Te\kern-1.00006ptV}$ data or
$\mbox{$p_{\mathrm{T}}$}>1.76\mathrm{\,Ge\kern-1.00006ptV}$ in the
8$\mathrm{\,Te\kern-1.00006ptV}$ data. In the subsequent software trigger, at
least one of the final-state particles is required to have
$\mbox{$p_{\mathrm{T}}$}>1.7\mathrm{\,Ge\kern-1.00006ptV}$ in the
7$\mathrm{\,Te\kern-1.00006ptV}$ data or
$\mbox{$p_{\mathrm{T}}$}>1.6\mathrm{\,Ge\kern-1.00006ptV}$ in the
8$\mathrm{\,Te\kern-1.00006ptV}$ data, unless the particle is identified as a
muon in which case $\mbox{$p_{\mathrm{T}}$}>1.0\mathrm{\,Ge\kern-1.00006ptV}$
is required. The final-state particles that satisfy these transverse momentum
criteria are also required to have an impact parameter larger than
100${\,\upmu\mathrm{m}}$ with respect to all of the primary $pp$ interaction
vertices (PVs) in the event. Finally, the tracks of two or more of the final-
state particles are required to form a vertex that is significantly displaced
from the PVs. In the subsequent offline selection, trigger signals are
required to be associated with reconstructed particles in the signal decay
chain.

Figure 1: Distribution of $m_{K^{+}K^{-}}$ near the $\phi$ peak before the
$\phi$ candidate selection. Non-$B^{+}$ backgrounds have been subtracted using
sPlot weights [36] obtained from a fit to the
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}K^{-}K^{+}}$
distribution. The default $\phi$ selection window is indicated with vertical
red lines. The fit (solid blue line) of a Breit–Wigner $\phi$ signal shape
plus two-body phase space function (dashed red line), convolved with a
Gaussian resolution function, is superimposed.

The offline data selection is very similar to that described in Ref. [21],
with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$
candidates required to satisfy the following criteria:
$\mbox{$p_{\mathrm{T}}$}(\mu)>0.55\mathrm{\,Ge\kern-1.00006ptV}$,
$\mbox{$p_{\mathrm{T}}$}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})>1.5\mathrm{\,Ge\kern-1.00006ptV}$, $\chi^{2}$ per degree of freedom
for the two muons to form a common vertex, $\chi^{2}_{\rm
vtx}(\mu^{+}\mu^{-})/\hbox{\rm ndf}<9$, and mass consistent with the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ meson. Every charged track
with $\mbox{$p_{\mathrm{T}}$}>0.25\mathrm{\,Ge\kern-1.00006ptV}$, missing all
PVs by at least 3 standard deviations ($\chi^{2}_{\rm IP}(K)>9$) and
classified as more likely to be a kaon than a pion according to the particle
identification system, is considered a kaon candidate. The quantity
$\chi^{2}_{\rm IP}(K)$ is defined as the difference between the $\chi^{2}$ of
the PV reconstructed with and without the considered particle. Combinations of
$K^{+}K^{-}K^{+}$ candidates that are consistent with originating from a
common vertex with $\chi^{2}_{\rm vtx}(K^{+}K^{-}K^{+})/\hbox{\rm ndf}<9$ are
selected. We combine ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
candidates with $K^{+}K^{-}K^{+}$ candidates to form $B^{+}$ candidates, which
must satisfy $\chi^{2}_{\rm vtx}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}K^{+})/\hbox{\rm ndf}<9$,
$\mbox{$p_{\mathrm{T}}$}(B^{+})>2\mathrm{\,Ge\kern-1.00006ptV}$ and have decay
time greater than $0.25$ ps. The ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}K^{+}$ mass is calculated using the known
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mass [37] and the $B^{+}$
vertex as constraints [38].

Four discriminating variables ($x_{i}$) are used in a likelihood ratio to
improve the background suppression: the minimal $\chi^{2}_{\rm IP}(K)$,
$\chi^{2}_{\rm vtx}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}K^{+})/\hbox{\rm ndf}$, $\chi^{2}_{\rm IP}(B^{+})$, and the
cosine of the largest opening angle between the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and the kaon transverse
momenta. The latter peaks at positive values for the signal as the $B^{+}$
meson has high transverse momentum. Background events in which particles are
combined from two different $B$ decays peak at negative values, whilst those
due to random combinations of particles are more uniformly distributed. The
four signal probability density functions (PDFs), $\mathcal{P}_{\rm
sig}(x_{i})$, are obtained from simulated
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decays. The background PDFs, $\mathcal{P}_{\rm bkg}(x_{i})$, are obtained from
candidates in data with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}K^{+}$ invariant mass between 5.6 and
$6.4\mathrm{\,Ge\kern-1.00006ptV}$. We require
$-2\sum_{i=1}^{4}\ln[\mathcal{P}_{\rm sig}(x_{i})/\mathcal{P}_{\rm
bkg}(x_{i})]<5$, which retains about 90% of the signal events.

Relative to the data selection described in Ref. [21], the requirements on
transverse momentum for $\mu$ and $B^{+}$ candidates have been lowered and the
requirement on the multivariate signal-to-background log-likelihood difference
was loosened. As a result, the $B^{+}$ signal yield per unit luminosity has
increased by about 50% at the expense of somewhat higher background.

Figure 2: Mass of $B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi K^{+}$ candidates in the data (black points with error bars)
together with the results of the fit (blue line) with a double-sided Crystal
Ball shape for the $B^{+}$ signal on top of a quadratic function for the
background (red dashed line). The fit is used to determine the background
fraction under the peak in the mass range used in the amplitude analysis
(indicated with vertical solid red lines). The sidebands used for the
background parameterization are indicated with vertical dashed blue lines.

The distribution of $m_{K^{+}K^{-}}$ for the selected
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}K^{+}$ candidates is shown in Fig. 1 (two entries per
candidate). A fit with a P-wave relativistic Breit–Wigner shape on top of a
two-body phase space distribution representing non-$\phi$ background, both
convolved with a Gaussian resolution function with width of
$1.2\mathrm{\,Me\kern-1.00006ptV}$ is superimposed. Integration of the fit
components gives $(5.3\pm 0.5)\%$ of nonresonant background in the
$|m_{K^{+}K^{-}}-1020~{}\mathrm{\,Me\kern-1.00006ptV}|<15\mathrm{\,Me\kern-1.00006ptV}$
region used to define a $\phi$ candidate. To avoid reconstruction ambiguities,
we require that there is exactly one $\phi$ candidate per
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}K^{-}K^{+}$ combination,
which reduces the $B^{+}$ yield by $3.2\%$. The non-$\phi$
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}K^{+}$ background in the remaining sample is small ($2.1\%$)
and neglected in the amplitude model. The related systematic uncertainty is
estimated by tightening the $\phi$ mass selection window to $\pm
7\mathrm{\,Me\kern-1.00006ptV}$.

The mass distribution of the remaining ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi K^{+}$ combinations is shown in Fig. 2 together with a fit of the
$B^{+}$ signal represented by a symmetric double-sided Crystal Ball function
[39] on top of a quadratic function for the background. The fit yields
$4289\pm 151$ $B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi K^{+}$ events. Integration of the fit components in the
$5270$–$5290\mathrm{\,Me\kern-1.00006ptV}$ region (twice the $B^{+}$ mass
resolution on each side of its peak) used in the amplitude fits, gives a
background fraction ($\beta$) of $(23\pm 6)\%$. A Gaussian signal shape and a
higher-order polynomial background function are used to assign systematic
uncertainties which are included in, and dominate, the uncertainty given
above. The $B^{+}$ invariant mass sidebands, $5225$–$5256$ and
$5304$–$5335\mathrm{\,Me\kern-1.00006ptV}$, are used to parameterize the
background in the amplitude fit.

The $B^{+}$ candidates for the amplitude analysis are kinematically
constrained to the known $B^{+}$ mass [37] and to point to the closest $pp$
interaction vertex. The measured value of $m_{K^{+}K^{-}}$ is used for the
$\phi$ candidate mass, since the natural width of the $\phi$ resonance is
larger than the detector resolution.

##  4 Matrix element model

We consider the three interfering processes corresponding to the following
decay sequences: $B^{+}\rightarrow
K^{*+}{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, $K^{*+}\rightarrow\phi
K^{+}$ (referred to as the ${{K}^{*}}$ decay chain), $B^{+}\rightarrow
XK^{+}$, $X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ ($X$
decay chain) and $B^{+}\rightarrow Z^{+}\phi$,
$Z^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ ($Z$
decay chain), all followed by ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow\mu^{+}\mu^{-}$ and $\phi\rightarrow K^{+}K^{-}$ decays.
Here, $K^{*+}$, $X$ and $Z^{+}$ should be understood as any $\phi K^{+}$,
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ and
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ contribution,
respectively.

Figure 3: Definition of the $\theta_{{{K}^{*}}}$,
$\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$, $\theta_{\phi}$,
$\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$ and
$\Delta{\phi}_{{{K}^{*}},\phi}$ angles describing angular correlations in
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*+}$,
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$,
$K^{*+}\rightarrow\phi K^{+}$, $\phi\rightarrow K^{+}K^{-}$ decays
(${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is denoted as $\psi$ in the
figure).

We construct a model of the matrix element ($\mathcal{M}$) using the helicity
formalism [40, 41, 42] in which the six independent variables fully describing
the $K^{*+}$ decay chain are $m_{\phi K}$, $\theta_{{{K}^{*}}}$,
$\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$, $\theta_{\phi}$,
$\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$ and
$\Delta{\phi}_{{{K}^{*}},\phi}$, where the helicity angle $\theta_{P}$ is
defined as the angle in the rest frame of $P$ between the momentum of its
decay product and the boost direction from the rest frame of the particle
which decays to $P$, and $\Delta\phi$ is the angle between the decay planes of
the two particles (see Fig. 3). The set of angles is denoted by $\Omega$. The
explicit formulae for calculation of the angles in $\Omega$ are given in
Appendix A.

The full six-dimensional (6D) matrix element for the ${{K}^{*}}$ decay chain
is given by

$\begin{split}\mathcal{M}^{{{K}^{*}}}_{\Delta\lambda_{\mu}}\equiv&{\sum_{j}}\,{\,}R_{j}(m_{\phi{K}}){\sum_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}=-1,0,1}}\,{\sum_{\lambda_{\phi}=-1,0,1}}\,\,{A_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}\,j}}{\,}{A_{\lambda_{\phi}}^{{{K}^{*}}\rightarrow\phi
K\,j}}{\,}\times\\\
&\quad\quad{d^{J_{{{K}^{*}}}\,j}_{\rm\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\lambda_{\phi}}(\theta_{{{K}^{*}}})}{\,}{d^{1}_{\rm\lambda_{\phi},0}(\theta_{\phi})}{\,}e^{i\lambda_{\phi}\Delta{\phi}_{{{K}^{*}},\phi}}{\,}{d^{1}_{\rm\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\Delta\lambda_{\mu}}(\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}})}{\,}e^{i\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}},\\\ |\mathcal{M}^{{{K}^{*}}}|^{2}=&{\sum_{\Delta\lambda_{\mu}=\pm
1}}\,\left|\mathcal{M}^{{{K}^{*}}}_{\Delta\lambda_{\mu}}\right|^{2},\end{split}$
(1)

where the index $j$ enumerates the different ${{K}^{*+}}$ resonances. The
symbol $J_{{K}^{*}}$ denotes the spin of the ${{K}^{*}}$ resonance, $\lambda$
is the helicity (projection of the particle spin onto its momentum in the rest
frame of its parent) and
$\Delta\lambda_{\mu}\equiv\lambda_{{\mu^{+}}}-\lambda_{\mu^{-}}$. The terms
$d^{J}_{\rm\lambda_{1},\lambda_{2}}(\theta)$ are the Wigner d-functions,
$R_{j}(m_{\phi{K}})$ is the mass dependence of the contribution and will be
discussed in more detail later (usually a complex Breit–Wigner amplitude
depending on resonance pole mass $M_{0\,{{K}^{*}}\,j}$ and width
$\Gamma_{0\,{{K}^{*}}\,j}$). The coefficients
${A_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}}$ and ${A_{\lambda_{\phi}}^{{{K}^{*}}\rightarrow\phi K}}$
are complex helicity couplings describing the (weak) $B^{+}$ and (strong)
$K^{*+}$ decay dynamics, respectively. There are three independent complex
${A_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}}$ couplings to be fitted
($\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}=-1,0,1$) per
${{K}^{*}}$ resonance, unless $J_{{K}^{*}}=0$ in which case there is only one
since $\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}=\lambda_{{K}^{*}}$ due to $J_{B}=0$. Parity conservation in the
${{K}^{*}}$ decay limits the number of independent helicity couplings
${A_{\lambda_{\phi}}^{{{K}^{*}}\rightarrow\phi K}}$. More generally parity
conservation requires

$A_{-\lambda_{B},-\lambda_{C}}^{A\rightarrow
B\,C}=P_{A}\,P_{B}\,P_{C}\,(-1)^{J_{B}+J_{C}-J_{A}}\,A_{\lambda_{B},\,\lambda_{C}}^{A\rightarrow
B\,C},$ (2)

which, for the decay ${{K}^{*+}}\rightarrow\phi K^{+}$, leads to

$A_{\lambda_{\phi}}={P_{{K}^{*}}}(-1)^{J_{{{K}^{*}}}+1}\,A_{-\lambda_{\phi}}.$
(3)

This reduces the number of independent couplings in the ${{K}^{*}}$ decay to
one or two. Since the overall magnitude and phase of these couplings can be
absorbed in ${A_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}}$, in practice the ${{K}^{*}}$ decay contributes zero or one
complex parameter to be fitted per ${{K}^{*}}$ resonance.

The matrix element for the $X$ decay chain can be parameterized using
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$ and the $\theta_{X}$,
$\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}^{X}$,
$\theta_{\phi}^{X}$, $\Delta\phi_{X,{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$, $\Delta\phi_{X,\phi}$ angles. The angles
$\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}^{X}$ and
$\theta_{\phi}^{X}$ are not the same as
$\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$ and $\theta_{\phi}$
in the $K^{*}$ decay chain, since ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ and $\phi$ are produced in decays of different particles. For the
same reason, the muon helicity states are different between the two decay
chains, and an azimuthal rotation by angle $\alpha^{X}$ is needed to align
them as discussed below. The parameters needed to characterize the $X$ decay
chain, including $\alpha^{X}$, do not constitute new degrees of freedom since
they can all be derived from $m_{\phi K}$ and $\Omega$. The matrix element for
the $X$ decay chain also has unique helicity couplings and is given by

$\begin{split}\mathcal{M}^{X}_{\Delta\lambda_{\mu}}\equiv&{\sum_{j}}\,{\,}R_{j}(m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}){\sum_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}=-1,0,1}}\,{\sum_{\lambda_{\phi}=-1,0,1}}\,\,A_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\lambda_{\phi}}^{X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi\,j}\,\times\\\
&\quad\quad{\,}{d^{J_{X}\,j}_{\rm{0,\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}-\lambda_{\phi}}}(\theta_{X})}{\,}{d^{1}_{\rm\lambda_{\phi},0}(\theta_{\phi}^{X})}{\,}e^{i\lambda_{\phi}\Delta\phi_{X,\phi}}{\,}{d^{1}_{\rm\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\Delta\lambda_{\mu}}(\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}^{X})}{\,}e^{i\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\Delta\phi_{X,{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}},\\\
|{\mathcal{M}^{K^{*}+X}}|^{2}=&{\sum_{\Delta\lambda_{\mu}=\pm
1}}\left|\mathcal{M}^{{{K}^{*}}}_{\Delta\lambda_{\mu}}+{e^{i\alpha^{X}\Delta\lambda_{\mu}}}\mathcal{M}^{X}_{\Delta\lambda_{\mu}}\right|^{2},\end{split}$
(4)

where the index $j$ enumerates all $X$ resonances. To add
$\mathcal{M}^{{{K}^{*}}}_{\Delta\lambda_{\mu}}$ and
$\mathcal{M}^{X}_{\Delta\lambda_{\mu}}$ coherently it is necessary to
introduce the ${e^{i\alpha^{X}\Delta\lambda_{\mu}}}$ term, which corresponds
to a rotation about the ${\mu^{+}}$ momentum axis by the angle $\alpha^{X}$ in
the rest frame of ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ after
arriving to it by a boost from the $X$ rest frame. This realigns the
coordinate axes for the muon helicity frame in the $X$ and ${{K}^{*}}$ decay
chains. This issue is discussed in Ref. [43] and at more length in Ref. [44].

The structure of helicity couplings in the $X$ decay chain is different from
the ${{K}^{*}}$ decay chain. The decay $B^{+}\rightarrow XK^{+}$ does not
contribute any helicity couplings to the fit333There is one additional
coupling, but that can be absorbed by a redefinition of $X$ decay couplings,
which are free parameters., since $X$ is produced fully polarized
$\lambda_{X}=0$. The $X$ decay contributes a resonance-dependent matrix of
helicity couplings $A_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\lambda_{\phi}}^{X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}$. Fortunately, parity conservation reduces the number of
independent complex couplings to one for $J^{P}_{X}=0^{-}$, two for $0^{+}$,
three for $1^{+}$, four for $1^{-}$ and $2^{-}$, and at most five independent
couplings for $2^{+}$.

The matrix element for the $Z^{+}$ decay chain can be parameterized using
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K}$ and the $\theta_{Z}$,
$\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}^{Z}$,
$\theta_{\phi}^{Z}$, $\Delta\phi_{Z,{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$, $\Delta\phi_{Z,\phi}$ angles. The $Z^{+}$ decay chain also requires
a rotation to align the muon frames to those used in the ${{K}^{*}}$ decay
chain and to allow for the proper description of interference between the
three decay chains. The full 6D matrix element is given by

$\begin{split}\mathcal{M}_{\Delta\lambda_{\mu}}^{Z}\equiv&{\sum_{j}}\,{\,}R_{j}(m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{K}}){\sum_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}=-1,0,1}}\,{\sum_{\lambda_{\phi}=-1,0,1}}\,A_{\lambda_{\phi}}^{B\rightarrow
Z\phi\,j}{\,}A_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}^{Z\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K\,j}\,\times\\\
&\quad\quad{\,}{d^{J_{Z\,j}}_{\rm\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}(\theta_{Z})}{\,}{d^{1}_{\rm\lambda_{\phi},0}(\theta_{\phi}^{Z})}{\,}e^{i\lambda_{\phi}{\Delta\phi}_{Z,\phi}}{\,}{d^{1}_{\rm\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\Delta\lambda_{\mu}}(\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}^{Z})}{\,}e^{i\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}},\\\
|\mathcal{M}^{{{K}^{*}}+X+Z}|^{2}=&{\sum_{\Delta\lambda_{\mu}=\pm
1}}\left|\mathcal{M}^{{{K}^{*}}}_{\Delta\lambda_{\mu}}+{e^{i\alpha^{X}\Delta\lambda_{\mu}}}\mathcal{M}^{X}_{\Delta\lambda_{\mu}}+{e^{i\alpha^{Z}\Delta\lambda_{\mu}}}\mathcal{M}^{Z}_{\Delta\lambda_{\mu}}\right|^{2}.\end{split}$
(5)

Parity conservation in the $Z^{+}$ decay requires

$A_{\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}}^{B\rightarrow
Z\phi}={P_{Z}}(-1)^{J_{Z}+1}\,A_{-\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}^{B\rightarrow Z\phi}$ (6)

and provides a similar reduction of the couplings as discussed for the
${{K}^{*}}$ decay chain.

Instead of fitting the helicity couplings
$A_{\lambda_{B},\lambda_{C}}^{A\rightarrow B\,C}$ as free parameters, after
imposing parity conservation for the strong decays, it is convenient to
express them by an equivalent number of independent $LS$ couplings ($B_{LS}$),
where $L$ is the orbital angular momentum in the decay and $S$ is the total
spin of $B$ and $C$, $\vec{S}=\vec{J}_{B}+\vec{J}_{C}$ ($|J_{B}-J_{C}|\leq
S\leq J_{B}+J_{C}$). Possible combinations of $L$ and $S$ values are
constrained via $\vec{J}_{A}=\vec{L}+\vec{S}$. The relation involves the
Clebsch-Gordan coefficients

${A_{\lambda_{B},\lambda_{C}}^{A\rightarrow
B\,C}}=\sum_{L}\sum_{S}\sqrt{\frac{2L+1}{2J_{A}+1}}\,B_{L,S}\,\left(\begin{array}[]{cc|c}J_{B}&J_{C}&S\\\
\lambda_{B}&-\lambda_{C}&\lambda_{B}-\lambda_{C}\end{array}\right)\,\left(\begin{array}[]{cc|c}L&S&J_{A}\\\
0&\lambda_{B}-\lambda_{C}&\lambda_{B}-\lambda_{C}\end{array}\right).$ (7)

Parity conservation in the strong decays is imposed by

$P_{A}=P_{B}\,P_{C}\,(-1)^{L}.$ (8)

Since the helicity or $LS$ couplings not only shape the angular distributions
but also describe the overall strength and phase of the given contribution
relative to all other contributions in the matrix element, we separate these
roles by always setting the coupling for the lowest $L$ and $S$, $B_{L_{\rm
min}S_{\rm min}}$, for a given contribution to $(1,0)$ and multiplying the sum
in Eq. (7) by a complex fit parameter ${\cal A}$ (this is equivalent to
factoring out $B_{L_{\rm min}S_{\rm min}}$). This has an advantage when
interpreting the numerical values of these parameters. The value of ${\cal
A}_{j}$ describes the relative magnitude and phase of the $B_{L_{\rm
min}S_{\rm min}\,j}$ to the other contributions, and the fitted $B_{LS\,j}$
values correspond to the ratios, $B_{LS\,j}/B_{L_{\rm min}S_{\rm min}\,j}$,
and determine the angular distributions.

Each contribution to the matrix element comes with its own $R(m_{A})$
function, which gives its dependence on the invariant mass of the intermediate
resonance $A$ in the decay chain ($A=K^{*+}$, $X$ or $Z^{+}$). Usually it is
given by the Breit–Wigner amplitude, but there are special cases which we
discuss below. An alternative parameterization of $R(m_{A})$ to represent
coupled-channel cusps is discussed in Appendix D.

In principle, the width of the $\phi$ resonance should also be taken into
account. However, since the $\phi$ resonance is very narrow
($\Gamma_{0}=4.3\mathrm{\,Me\kern-1.00006ptV}$, with mass resolution of
$1.2\mathrm{\,Me\kern-1.00006ptV}$) we omit the amplitude dependence on the
invariant $m_{K^{+}K^{-}}$ mass from the $\phi$ decay.

A single resonant contribution in the decay chain $B^{+}\rightarrow A...$,
$A\rightarrow...$ is parameterized by the relativistic Breit–Wigner amplitude
together with Blatt–Weisskopf functions,

$R(m|M_{0},\Gamma_{0})=B^{\prime}_{L_{B}}(p,p_{0},d)\left(\frac{p}{p_{0}}\right)^{L_{B}}\,BW(m|M_{0},\Gamma_{0})\,B^{\prime}_{L_{A}}(q,q_{0},d)\left(\frac{q}{q_{0}}\right)^{L_{A}},$
(9)

where

$BW(m|M_{0},\Gamma_{0})=\frac{1}{M_{0}^{2}-m^{2}-iM_{0}\Gamma(m)}\,,$ (10)

is the Breit–Wigner amplitude including the mass-dependent width,

$\Gamma(m)=\Gamma_{0}\left(\frac{q}{q_{0}}\right)^{2\,L_{A}+1}\frac{M_{0}}{m}B^{\prime}_{L_{A}}(q,q_{0},d)^{2}\,.$
(11)

Here, $p$ is the momentum of the resonance $A$ ($K^{*+}$, $X$ or $Z^{+}$) in
the $B^{+}$ rest frame, and $q$ is the momentum of one of the decay products
of $A$ in the rest frame of the $A$ resonance. The symbols $p_{0}$ and $q_{0}$
are used to indicate values of these quantities at the resonance peak mass
($m=M_{0}$). The orbital angular momentum in $B^{+}$ decay is denoted as
$L_{B}$, and that in the decay of the resonance $A$ as $L_{A}$. The orbital
angular momentum barrier factors, $p^{L}\,B^{\prime}_{L}(p,p_{0},d)$, involve
the Blatt–Weisskopf functions [45, 46]:

$\displaystyle B^{\prime}_{0}(p,p_{0},d)$ $\displaystyle=$ $\displaystyle
1\,,$ (12) $\displaystyle B^{\prime}_{1}(p,p_{0},d)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1+(p_{0}\,d)^{2}}{1+(p\phantom{{}_{0}}\,d)^{2}}}\,,$
(13) $\displaystyle B^{\prime}_{2}(p,p_{0},d)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{9+3(p_{0}\,d)^{2}+(p_{0}\,d)^{4}}{9+3(p\phantom{{}_{0}}\,d)^{2}+(p\phantom{{}_{0}}\,d)^{4}}}\,,$
(14) $\displaystyle B^{\prime}_{3}(p,p_{0},d)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{225+45(p_{0}\,d)^{2}+6(p_{0}\,d)^{4}+(p_{0}\,d)^{6}}{225+45(p\phantom{{}_{0}}\,d)^{2}+6(p\phantom{{}_{0}}\,d)^{4}+(p\phantom{{}_{0}}\,d)^{6}}}\,,$
(15) $\displaystyle B^{\prime}_{4}(p,p_{0},d)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{11025+1575(p_{0}\,d)^{2}+135(p_{0}\,d)^{4}+10(p_{0}\,d)^{6}+(p_{0}\,d)^{8}}{11025+1575(p\phantom{{}_{0}}\,d)^{2}+135(p\phantom{{}_{0}}\,d)^{4}+10(p\phantom{{}_{0}}\,d)^{6}+(p\phantom{{}_{0}}\,d)^{8}}}\,,$
(16)

which account for the centrifugal barrier in the decay and depend on the
momentum of the decay products in the rest frame of the decaying particle
($p$) as well as the size of the decaying particle ($d$). In this analysis we
set this parameter to a nominal value of
$d=3.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}$, and vary it in between $1.5$ and
$5.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}$ in the evaluation of the systematic
uncertainty.

In the helicity approach, each helicity state is a mixture of many different
$L$ values. We follow the usual approach of using in Eq. (9) the minimal
$L_{B}$ and $L_{A}$ values allowed by the quantum numbers of the given
resonance $A$, while higher values are used to estimate the systematic
uncertainty.

We set $BW(m)=1.0$ for the nonresonant (NR) contributions, which means
assuming that both magnitude and phase have negligible $m$ dependence. As the
available phase space in the
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decays is small (the energy release is only 12% of the $B^{+}$ mass) this is a
well-justified assumption. We consider possible mass dependence of NR
amplitudes as a source of systematic uncertainties.

##  5 Maximum likelihood fit of amplitude models

The signal PDF, $\mathcal{P}_{\rm sig}$, is proportional to the matrix element
squared, which is a function of six independent variables: $m_{\phi K}$ and
the independent angular variables in the ${{K}^{*}}$ decay chain $\Omega$. The
PDF also depends on the fit parameters, $\vec{\omega}$, which include the
helicity couplings, and masses and widths of resonances. The two other
invariant masses, $m_{\phi K}$ and $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K}$, and the angular variables describing the $X$ and $Z^{+}$ decay
chains depend on $m_{\phi K}$ and $\Omega$, therefore they do not represent
independent dimensions. The signal PDF is given by:

$\frac{d\mathcal{P}}{dm_{\phi K}\,d\Omega}\equiv\mathcal{P}_{\rm sig}(m_{\phi
K},\Omega|\vec{\omega})=\frac{1}{I(\overrightarrow{w})}\big{|}\mathcal{M}(m_{\phi
K},\Omega|\vec{\omega})\big{|}^{2}\Phi(m_{\phi K})\epsilon(m_{\phi
K},\Omega),$ (17)

where $\mathcal{M}(m_{\phi K},\Omega|\vec{\omega})$ is the matrix element
given by Eq. (5). $\Phi(m_{\phi K})=p\,q$ is the phase space function, where
$p$ is the momentum of the $\phi K^{+}$ (i.e. ${{K}^{*}}$) system in the
$B^{+}$ rest frame, and $q$ is the $K^{+}$ momentum in the $K^{*+}$ rest
frame. The function $\epsilon(m_{\phi K},\Omega)$ is the signal efficiency,
and $I(\vec{\omega})$ is the normalization integral,

$I(\vec{\omega})\equiv\int\mathcal{P}_{\rm sig}(m_{\phi K},\Omega)\,dm_{\phi
K}\,d\Omega\propto\frac{\sum_{j}w_{j}^{\rm
MC}\big{|}\mathcal{M}(m_{Kp~{}j},\Omega_{j}|\vec{\omega})\big{|}^{2}}{\sum_{j}w_{j}^{\rm
MC}},$ (18)

where the sum is over simulated events, which are generated uniformly in
$B^{+}$ decay phase space and passed through the detector simulation [47] and
data selection. In the simulation, $pp$ collisions producing $B^{+}$ mesons
are generated using Pythia [48] with a specific LHCb configuration [49]. The
weights $w_{j}^{\rm MC}$ introduced in Eq. (18) contain corrections to the
$B^{+}$ production kinematics in the generation and to the detector response
to bring the simulations into better agreement with the data. Setting
$w_{j}^{\rm MC}=1$ is one of the variations considered when evaluating
systematic uncertainties. The simulation sample contains 132 000 events,
approximately 30 times the signal size in data. This procedure folds the
detector response into the model and allows a direct determination of the
parameters of interest from the uncorrected data. The resulting log-likelihood
sums over the data events (here for illustration,
$\mathcal{P}=\mathcal{P}_{\rm sig}$),

$\begin{split}\ln L(\vec{\omega})&=\sum_{i}\ln\mathcal{P}_{\rm
sig}(m_{Kp~{}i},\Omega_{i}|\vec{\omega})\\\
&=\sum_{i}\ln\big{|}\mathcal{M}(m_{Kp~{}i},\Omega_{i}|\vec{\omega})\big{|}^{2}-N\ln
I(\vec{\omega})+\sum_{i}\ln[\Phi(m_{Kp~{}i})\epsilon(m_{Kp~{}i},\Omega_{i})],\end{split}$
(19)

where the last term does not depend on $\vec{\omega}$ and can be dropped ($N$
is the total number of the events in the fit).

In addition to the signal PDF, $\mathcal{P}_{\rm sig}(m_{\phi
K},\Omega|\vec{\omega})$, the background PDF, $\mathcal{P}_{\rm bkg}(m_{\phi
K},\Omega)$ determined from the $B^{+}$ mass peak sidebands, is included. We
minimize the negative log-likelihood defined as

$\begin{split}&-\ln
L(\vec{\omega})=-\sum_{i}\ln\left[(1-\beta)\,\mathcal{P}_{\rm sig}(m_{\phi
K~{}i},\Omega_{i}|\vec{\omega})+\beta\,\mathcal{P}_{\rm bkg}(m_{\phi
K~{}i},\Omega_{i})\right]\\\
&=-\sum_{i}\ln\left[(1-\beta)\,\frac{\big{|}\mathcal{M}(m_{\phi
K~{}i},\Omega_{i}|\vec{\omega})\big{|}^{2}\Phi(m_{\phi K~{}i})\epsilon(m_{\phi
K~{}i},\Omega_{i})}{I(\vec{\omega})}+\beta\,\frac{\mathcal{P}_{\rm
bkg}^{u}(m_{\phi K~{}i},\Omega_{i})}{I_{\rm bkg}}\right]\\\
&=-\sum_{i}\ln\left[\big{|}\mathcal{M}(m_{\phi
K~{}i},\Omega_{i}|\vec{\omega})\big{|}^{2}+\frac{\beta\,I(\vec{\omega})}{(1-\beta)I_{\rm
bkg}}\,\frac{\mathcal{P}_{\rm bkg}^{u}(m_{\phi
K~{}i},\Omega_{i})}{\Phi(m_{\phi K~{}i})\epsilon(m_{\phi
K~{}i},\Omega_{i})}\right]+N\ln I(\vec{\omega})+{\rm const.},\end{split}$ (20)

where $\beta$ is the background fraction in the peak region determined from
the fit to the $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K}$
distribution (Fig. 2), $\mathcal{P}_{\rm bkg}^{u}(m_{\phi K},\Omega)$ is the
unnormalized background density proportional to the density of sideband
events, with its normalization determined by444Notice that the distribution of
MC events includes both the $\Phi(m_{\phi K})$ and $\epsilon(m_{\phi
K},\Omega)$ factors, which cancel their product in the numerator.

$I_{\rm bkg}\equiv\int\mathcal{P}_{\rm bkg}^{u}(m_{\phi K})\,dm_{\phi
K}\,d\Omega\,\propto\,\frac{\sum_{j}w_{j}^{\rm MC}\frac{\mathcal{P}_{\rm
bkg}^{u}(m_{\phi K~{}j},\Omega_{j})}{\Phi(m_{\phi K~{}i})\epsilon(m_{\phi
K~{}j},\Omega_{j})}}{\sum_{j}w_{j}^{\rm MC}}.$ (21)

The equation above implies that the background term is efficiency corrected,
so it can be added to the efficiency-independent signal probability expressed
by $\left|\mathcal{M}\right|^{2}$. This way the efficiency parameterization,
$\epsilon(m_{\phi K},\Omega)$, becomes a part of the background description
which affects only a small part of the total PDF.

The efficiency parameterization in the background term is assumed to factorize
as

$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\epsilon(m_{\phi
K},\Omega)=\epsilon_{1}(m_{\phi
K},\cos\theta_{{{K}^{*}}})\,\,\epsilon_{2}(\cos\theta_{\phi}|m_{\phi
K})\times$\mbox{}\hfil\\\
&\quad\quad\quad\quad\epsilon_{3}(\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}|m_{\phi K})\,\,\epsilon_{4}(\Delta{\phi}_{{{K}^{*}},\phi}|m_{\phi
K})\,\,\epsilon_{5}(\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}|m_{\phi K}).\end{split}$ (22)

The $\epsilon_{1}(m_{\phi K},\cos\theta_{{{K}^{*}}})$ term is obtained by
binning a two-dimensional (2D) histogram of the simulated signal events. Each
event is given a $1/(p\,q)$ weight, since at the generator level the phase
space is flat in $\cos\theta_{{{K}^{*}}}$ but has a $p\,q$ dependence on
$m_{\phi K}$. A bi-cubic function is used to interpolate between bin centers.
The $\epsilon_{1}(m_{\phi K},\cos\theta_{{{K}^{*}}})$ efficiency and its
visualization across the normal Dalitz plane are shown in Fig. 4. The other
terms are again built from 2D histograms, but with each bin divided by the
number of simulated events in the corresponding $m_{\phi K}$ slice to remove
the dependence on this mass (Fig. 5).

Figure 4: Parameterized efficiency $\epsilon_{1}(m_{\phi
K},\cos\theta_{{{K}^{*}}})$ function (top) and its representation in the
Dalitz plane $({m_{\phi{K}}^{2}},{m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}^{2}})$ (bottom). Function values corresponding to the color
encoding are given on the right. The normalization arbitrarily corresponds to
unity when averaged over the phase space.  Figure 5: Parameterized efficiency
$\epsilon_{2}(\cos\theta_{\phi}|m_{\phi K})$,
$\epsilon_{3}(\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}|m_{\phi K})$, $\epsilon_{4}(\Delta{\phi}_{{{K}^{*}},\phi}|m_{\phi
K})$,
$\epsilon_{5}(\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}|m_{\phi K})$ functions. Function values corresponding to the color
encoding are given on the right. By construction each function integrates to
unity at each $m_{\phi K}$ value. The structure in
$\epsilon_{2}(\cos\theta_{\phi}|m_{\phi K})$ present between 1500 and
$1600\mathrm{\,Me\kern-0.90005ptV}$ is an artifact of removing
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}K^{+}$ events in which both $K^{+}K^{-}$ combinations pass
the $\phi$ mass selection window.

The background PDF, $\mathcal{P}_{\rm bkg}^{u}(m_{\phi K},\Omega)/\Phi(m_{\phi
K})$, is built using the same approach,

$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\frac{{\mathcal{P}_{\rm
bkg}^{u}}(m_{\phi K},\Omega)}{\Phi(m_{\phi K})}={P_{\rm bkg}}_{1}(m_{\phi
K},\cos\theta_{{{K}^{*}}})\,\,{P_{\rm bkg}}_{2}(\cos\theta_{\phi}|m_{\phi
K})\times$\mbox{}\hfil\\\ &\quad\quad\quad{P_{\rm
bkg}}_{3}(\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}|m_{\phi
K})\,\,{P_{\rm bkg}}_{4}(\Delta{\phi}_{{{K}^{*}},\phi}|m_{\phi K})\,\,{P_{\rm
bkg}}_{5}(\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}|m_{\phi K}).\end{split}$ (23)

The background function ${P_{\rm bkg}}_{1}(m_{\phi K},\cos\theta_{{K}^{*}})$
is shown in Fig. 6 and the other terms are shown in Fig. 7.

Figure 6: Parameterized background ${P_{\rm bkg}}_{1}(m_{\phi
K},\cos\theta_{{{K}^{*}}})$ function (top) and its representation in the
Dalitz plane $({m_{\phi{K}}^{2}},{m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}^{2}})$ (bottom). Function values corresponding to the color
encoding are given on the right. The normalization arbitrarily corresponds to
unity when averaged over the phase space.  Figure 7: Parameterized background
functions: $P^{u}_{\rm bkg\,2}(\cos\theta_{\phi}|m_{\phi{K}})$, $P^{u}_{\rm
bkg\,3}(\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}|m_{\phi{K}})$, $P^{u}_{\rm
bkg\,4}(\Delta{\phi}_{{{K}^{*}},\phi}|m_{\phi{K}})$, $P^{u}_{\rm
bkg\,5}(\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}|m_{\phi{K}})$. Function values corresponding to the color encoding
are given on the right. By construction each function integrates to unity at
each $m_{\phi K}$ value.

The fit fraction (FF) of any component $R$ is defined as,

${\rm FF}=\frac{\int\left|\mathcal{M}^{R}(m_{\phi
K},\Omega)\right|^{2}\Phi(m_{\phi K})\,dm_{\phi
K}d\Omega}{\int\left|\mathcal{M}(m_{\phi K},\Omega)\right|^{2}\Phi(m_{\phi
K})\,dm_{\phi K}d\Omega},$ (24)

where in $\mathcal{M}^{R}$ all terms except those associated with the $R$
amplitude are set to zero.

##  6 Background-subtracted and efficiency-corrected distributions

The background-subtracted and efficiency-corrected Dalitz plots are shown in
Figs. 8–10 and the mass projections are shown in Figs. 11–13. The latter
indicates that the efficiency corrections are rather minor. The background is
eliminated by subtracting the scaled $B^{+}$ sideband distributions. The
efficiency corrections are achieved by weighting events according to the
inverse of the parameterized 6D efficiency given by Eq. (22). The efficiency-
corrected signal yield remains similar to the signal candidate count, because
we normalize the efficiency to unity when averaged over the phase space.

Figure 8: Background-subtracted and efficiency-corrected data yield in the
Dalitz plane of $({m_{\phi{K}}^{2}},{m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}^{2}})$. Yield values corresponding to the color encoding are
given on the right.  Figure 9: Background-subtracted and efficiency-corrected
data yield in the Dalitz plane of
$({m_{\phi{K}}^{2}},{m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{K}}^{2}})$. Yield values corresponding to the color encoding are given
on the right.  Figure 10: Background-subtracted and efficiency-corrected data
yield in the Dalitz plane of $({m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{K}}^{2}},{m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}^{2}})$. Yield values corresponding to the color encoding are
given on the right.

While the $m_{\phi K}$ distribution (Fig. 11) does not contain any obvious
resonance peaks, it would be premature to conclude that there are none since
all $K^{*+}$ resonances expected in this mass range belong to higher
excitations, and therefore should be broad. In fact the narrowest known
$K^{*+}$ resonance in this mass range has a width of approximately
$150\mathrm{\,Me\kern-1.00006ptV}$ [37]. Scattering experiments sensitive to
$K^{*}\rightarrow\phi K$ decays also showed a smooth mass distribution, which
revealed some resonant activity only after partial-wave analysis [50, 51, 52].
Therefore, studies of angular distributions in correlation with $m_{\phi K}$
are necessary. Using full 6D correlations results in the best sensitivity.

The $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$ distribution
(Fig. 12) contains several peaking structures, which could be exotic or could
be reflections of conventional $K^{*+}$ resonances. There is no narrow
$X(4140)$ peak just above the kinematic threshold, consistent with the LHCb
analysis presented in Ref. [21], however we observe a broad enhancement. A
peaking structure is observed at about $4300\mathrm{\,Me\kern-1.00006ptV}$.
The high mass region is inspected with good sensitivity for the first time,
with the rate having a minimum near $4640\mathrm{\,Me\kern-1.00006ptV}$ with
two broad peaks on each side.

The $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K}$ distribution (Fig.
13) peaks broadly in the middle and has a high-mass peak, which is strongly
correlated with the low-mass $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}$ enhancement (Fig. 10).

As explained in the previous section, the amplitude fits are performed by
maximizing the unbinned likelihood on the selected signal candidates including
background events and without the efficiency weights. To properly represent
the fit quality, the fit projections in the following sections show the fitted
data sample, i.e. including the background and without the parameterized
efficiency corrections applied to the signal events.

Figure 11: Background-subtracted (histogram) and efficiency-corrected (points)
distribution of $m_{\phi K}$. See the text for the explanation of the
efficiency normalization.  Figure 12: Background-subtracted (histogram) and
efficiency-corrected (points) distribution of
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$. See the text for the
explanation of the efficiency normalization.  Figure 13: Background-subtracted
(histogram) and efficiency-corrected (points) distribution of
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K}$. See the text for the
explanation of the efficiency normalization.

##  7 Amplitude model with only $\phi K^{+}$ contributions

We first try to describe the data with kaon excitations alone. Their mass
spectrum as predicted in the relativistic potential model by Godfrey–Isgur
[53] is shown in Fig. 14 together with the experimentally determined masses of
both well-established and unconfirmed $K^{*}$ resonances [37]. Past
experiments on $K^{*}$ states decaying to $\phi K$ [50, 51, 52] had limited
precision, especially at high masses, gave somewhat inconsistent results, and
provided evidence for only a few of the states expected from the quark model
in the $1513$–$2182\mathrm{\,Me\kern-1.00006ptV}$ range probed in our data
set. However, except for the $J^{P}=0^{+}$ states which cannot decay to $\phi
K$ because of angular momentum and parity conservation, all other kaon
excitations above the $\phi K$ threshold are predicted to decay to this final
state [54]. In $B^{+}$ decays, production of high spin states, like the
$K^{*}_{3}(1780)$ or $K^{*}_{4}(2045)$ resonances, is expected to be
suppressed by the high orbital angular momentum required to produce them.

Figure 14: Kaon excitations predicted by Godfrey–Isgur [53] (horizontal black
lines) labeled with their intrinsic quantum numbers: $n{}^{2S+1}L_{J}$ (see
the text). Well established states are shown with narrower solid blue boxes
extending to $\pm 1\sigma$ in mass and labeled with their PDG names [37].
Unconfirmed states are shown with dashed green boxes. The long horizontal red
lines indicate the $\phi K$ mass range probed in
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decays.

We have used the predictions of the Godfrey–Isgur model as a guide to the
quantum numbers of the $K^{*+}$ states to be included in the model. The masses
and widths of all states are left free; thus our fits do not depend on
detailed predictions of Ref. [53], nor on previous measurements. We also allow
a constant nonresonant amplitude with $J^{P}=1^{+}$, since such $\phi K^{+}$
contributions can be produced, and can decay, in S-wave. Allowing the
magnitude of the nonresonant amplitude to vary with $m_{\phi K}$ does not
improve fit qualities.

While it is possible to describe the $m_{\phi K}$ and
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K}$ distributions well with
${{K}^{*}}$ contributions alone, the fit projections onto
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$ do not provide an
acceptable description of the data. For illustration we show in Fig. 15 the
projection of a fit with the following composition: a nonresonant term plus
candidates for two $2{\rm P}_{1}$, two $1{\rm D}_{2}$, and one of each of
$1{}^{3}{\rm F}_{3}$, $1{}^{3}{\rm D}_{1}$, $3{}^{3}{\rm S}_{1}$, $3{}^{1}{\rm
S}_{0}$, $2{}^{3}{\rm P}_{2}$, $1{}^{3}{\rm F}_{2}$, $1{}^{3}{\rm D}_{3}$ and
$1{}^{3}{\rm F}_{4}$ states, labeled here with their intrinsic quantum
numbers: $n{}^{2S+1}L_{J}$ ($n$ is the radial quantum number, $S$ the total
spin of the valence quarks, $L$ the orbital angular momentum between quarks,
and $J$ the total angular momentum of the bound state). The fit contains 104
free parameters. The $\chi^{2}$ value (144.9/68 bins) between the fit
projection and the observed $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}$ distribution corresponds to a p value below $10^{-7}$. Adding
more resonances does not change the conclusion that non-${{K}^{*}}$
contributions are needed to describe the data.

Figure 15: Distribution of $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}$ for the data and the fit results with a model containing only
$K^{*+}\rightarrow\phi K^{+}$ contributions.

##  8 Amplitude model with $\phi K^{+}$ and
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ contributions

We have explored adding $X$ and $Z^{+}$ contributions of various quantum
numbers to the fit models. Only $X$ contributions lead to significant
improvements in the description of the data. The default resonance model is
described in detail below and is summarized in Table 3, where the results are
also compared with the previous measurements and the theoretical predictions
for $\bar{s}u$ states [53]. The model contains seven $K^{*+}$ states, four $X$
states and $\phi K^{+}$ and ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ nonresonant components. There are 98 free parameters in this fit.
Projections of the fit onto the mass variables are displayed in Fig. 16. The
$\chi^{2}$ value (71.5/68 bins) between the fit projection and the observed
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$ distribution
corresponds to a p value of 22%. Projections onto angular variables are shown
in Figs. 17–19. Projections onto masses in different regions of the Dalitz
plot can be found in Fig. 20. Using adaptive binning555The adaptive binning
procedure maintains uniform and adequate bin contents. on the Dalitz plane
$m_{\phi K}^{2}$ vs. $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}^{2}$ (or extending the binning to all six fitted dimensions) the
$\chi^{2}$ value of 438.7/496 bins (462.9/501 bins) gives a p value of 17%
(2.3%). The $\chi^{2}$ PDFs used to obtain the p values have been obtained
with simulations of pseudoexperiments generated from the default amplitude
model.

The systematic uncertainties are obtained from the sum in quadrature of the
changes observed in the fit results when: the $K^{*+}$ and $X(4140)$ models
are varied; the Breit–Wigner amplitude parameterization is modified; only the
left or right $B^{+}$ mass peak sidebands are used for the background
parameterization; the $\phi$ mass selection window is made narrower by a
factor of two (to reduce the non-$\phi$ background fraction); the signal and
background shapes are varied in the fit to
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K}$ which determines
the background fraction $\beta$; the weights assigned to simulated events, in
order to improve agreement with the data on $B^{+}$ production characteristics
and detector efficiency, are removed. More detailed discussion of the
systematic uncertainties can be found in Appendix B.

The significance of each (non)resonant contribution is calculated assuming
that $\Delta(\\!-2\\!\ln{\cal L})$, after the contribution is included in the
fit, follows a $\chi^{2}$ distribution with the number of degrees of freedom
(${\rm ndf}$) equal to the number of free parameters in its parameterization.
The value of ${\rm ndf}$ is doubled when $M_{0}$ and $\Gamma_{0}$ are free
parameters in the fit. The validity of this assumption has been verified using
simulated pseudoexperiments. The significances of the $X$ contributions are
given after accounting for systematic variations. Combined significances of
exotic contributions, determined by removing more than one exotic contribution
at a time, are much larger than their individual significances given in Table
3. The significance of the spin-parity determination for each $X$ state is
determined as described in Appendix C.

The longitudinal ($f_{L}$) and transverse ($f_{\perp}$) polarizations are
calculated for $K^{*+}$ contributions according to

$f_{L}=\frac{\left|A_{\lambda={0}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}\right|^{2}}{\left|A_{\lambda={-1}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}\right|^{2}+\left|A_{\lambda={0}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}\right|^{2}+\left|A_{\lambda={+1}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}\right|^{2}},\\\ $ (25)
$f_{\perp}=\frac{\left|A_{{\perp}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}\right|^{2}}{\left|A_{\lambda={-1}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}\right|^{2}+\left|A_{\lambda={0}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}\right|^{2}+\left|A_{\lambda={+1}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}\right|^{2}},$ (26)

where

$A_{{\perp}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}=\frac{A_{\lambda={+1}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}-A_{\lambda={-1}}^{B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}}}{\sqrt{2}}.$ (27)

Table 3: Results for significances, masses, widths and fit fractions of the
components included in the default amplitude model. The first (second) errors
are statistical (systematic). Errors on $f_{L}$ and $f_{\perp}$ are
statistical only. Possible interpretations in terms of kaon excitation levels
are given, with notation $n{}^{2S+1}L_{J}$, together with the masses predicted
in the Godfrey-Isgur model [53]. Comparisons with the previously
experimentally observed kaon excitations [37] and
$X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ structures
are also given.

Contri- sign. Fit results bution or Ref. $M_{0}$
[$\mathrm{\,Me\kern-1.00006ptV}$ ] $\Gamma_{0}$
[$\mathrm{\,Me\kern-1.00006ptV}$ ] FF % $f_{L}$ $f_{\perp}$ All $K(1^{+})$
$8.0\sigma$ $42\\!\pm\\!\phantom{8}8\,^{+\phantom{8}5}_{-\phantom{8}9}$ ${\rm
NR}_{\phi K}$ $16\\!\pm\\!13\,^{+35}_{-\phantom{8}6}$ $0.52\pm 0.29$ $0.21\pm
0.16$ $K(1^{+})$ $7.6\sigma$ $1793\\!\pm\\!59\,^{+153}_{-101}$
$365\\!\pm\\!157\,^{+138}_{-215}$ $12\\!\pm\\!10\,^{+17}_{-\phantom{8}6}$
$0.24\pm 0.21$ $0.37\pm 0.17$ $2{}^{1}{\rm P}_{1}$ [53] $1900$ $K_{1}(1650)$
[37] $1650\\!\pm\\!50$ $150\\!\pm\\!\phantom{2}50$ $K^{{}^{\prime}}(1^{+})$
$1.9\sigma$ $1968\\!\pm\\!65\,^{+\phantom{8}70}_{-172}$
$396\\!\pm\\!170\,^{+174}_{-178}$ $23\\!\pm\\!20\,^{+31}_{-29}$ $0.04\pm 0.08$
$0.49\pm 0.10$ $2{}^{3}{\rm P}_{1}$ [53] $1930$ All $K(2^{-})$ $5.6\sigma$
$11\\!\pm\\!\phantom{8}3\,^{+\phantom{8}2}_{-\phantom{8}5}$ $K(2^{-})$
$5.0\sigma$ $1777\\!\pm\\!35\,^{+122}_{-\phantom{8}77}$
$217\\!\pm\\!116\,^{+221}_{-154}$ $0.64\pm 0.11$ $0.13\pm 0.13$ $1{}^{1}{\rm
D}_{2}$ [53] $1780$ ${\small K_{2}(1770)}$ [37] $1773\\!\pm\\!\phantom{2}8$
$188\\!\pm\\!\phantom{2}14$ $K^{{}^{\prime}}(2^{-})$ $3.0\sigma$
$1853\\!\pm\\!27\,^{+\phantom{2}18}_{-\phantom{2}35}$
$167\\!\pm\\!\phantom{2}58\,^{+\phantom{2}83}_{-\phantom{2}72}$ $0.53\pm 0.14$
$0.04\pm 0.08$ $1{}^{3}{\rm D}_{2}$ [53] $1810$ ${\small K_{2}(1820)}$ [37]
$1816\\!\pm\\!13$ $276\\!\pm\\!\phantom{2}35$ $K^{*}(1^{-})$ $8.5\sigma$
$1722\\!\pm\\!20\,^{+\phantom{8}33}_{-109}$
$354\\!\pm\\!\phantom{2}75\,^{+140}_{-181}$ $6.7\\!\pm\\!1.9\,^{+3.2}_{-3.9}$
$0.82\pm 0.04$ $0.03\pm 0.03$ $1{}^{3}{\rm D}_{1}$ [53] $1780$ ${\small
K^{*}(1680)}$ [37] $1717\\!\pm\\!27$ $322\\!\pm\\!110$ $K^{*}(2^{+})$
$5.4\sigma$ $2073\\!\pm\\!94\,^{+245}_{-240}$
$678\\!\pm\\!311\,^{+1153}_{-\phantom{2}559}$
$2.9\\!\pm\\!0.8\,^{+1.7}_{-0.7}$ $0.15\pm 0.06$ $0.79\pm 0.08$ $2{}^{3}{\rm
P}_{2}$ [53] $1940$ ${K^{*}_{2}(1980)}$ [37] $1973\\!\pm\\!26$
$373\\!\pm\\!\phantom{2}69$ $K(0^{-})$ $3.5\sigma$
$1874\\!\pm\\!43\,^{+\phantom{2}59}_{-115}$
$168\\!\pm\\!\phantom{2}90\,^{+280}_{-104}$ $2.6\\!\pm\\!1.1\,^{+2.3}_{-1.8}$
$1.0\phantom{5\pm 0.55}$ $3{}^{1}{\rm S}_{0}$ [53] $2020$ ${K(1830)}$ [37]
$\sim 1830$ $\sim 250\phantom{\pm 888}$ All $X(1^{+})$
$16\\!\pm\\!3\phantom{2}\phantom{2}\,^{+\phantom{2}6}_{-\phantom{2}2}$
$X(4140)$ $8.4\sigma$ $4146.5\\!\pm\\!4.5\,^{+4.6}_{-2.8}$
$83\\!\pm\\!21\,^{+21}_{-14}$ $13.0\\!\pm\\!3.2\,^{+4.8}_{-2.0}$ ave.
Table 1 $4147.1\\!\pm\\!2.4$ $15.7\\!\pm\\!6.3$ $X(4274)$ $6.0\sigma$
$4273.3\\!\pm\\!8.3\,^{+17.2}_{-\phantom{2}3.6}$
$56\\!\pm\\!11\,^{+\phantom{2}8}_{-11}$ $7.1\\!\pm\\!2.5\,^{+3.5}_{-2.4}$ CDF
[29] $4274.4\,^{+8.4}_{-6.7}\pm 1.9$ $32\,^{+22}_{-15}\pm 8$ CMS [25]
$4313.8\\!\pm\\!5.3\\!\pm\\!7.3$ $38\,^{+30}_{-15}\pm 16$ All $X(0^{+})$
$28\\!\pm\\!\phantom{2}5\\!\pm\\!\phantom{2}7$ ${\rm
NR}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$ $6.4\sigma$
$46\\!\pm\\!11\phantom{2}\,^{+11}_{-21}$ $X(4500)$ $6.1\sigma$
$4506\\!\pm\\!11\,^{+12}_{-15}$ $\phantom{8}92\\!\pm\\!21\,^{+21}_{-20}$
$6.6\\!\pm\\!2.4\,^{+3.5}_{-2.3}$ $X(4700)$ $5.6\sigma$
$4704\\!\pm\\!10\,^{+14}_{-24}$ $120\\!\pm\\!31\,^{+42}_{-33}$
$12\\!\pm\\!\phantom{2}5\phantom{2}\,^{+\phantom{2}9}_{-\phantom{2}5}$

Among the $K^{*+}$ states, the $J^{P}=1^{+}$ partial wave has the largest
total fit fraction (given by Eq. (24)). We describe it with three heavily
interfering contributions: a nonresonant term and two resonances. The
significance of the nonresonant amplitude cannot be quantified, since when it
is removed one of the resonances becomes very broad, taking over its role.
Evidence for the first $1^{+}$ resonance is significant ($7.6\sigma$). We
include a second resonance in the model, even though it is not significant
($1.9\sigma$), because two states are expected in the quark model. We remove
it as a systematic variation. The $1^{+}$ states included in our model appear
in the mass range where two $2{\rm P}_{1}$ states are predicted (see Table 3),
and where the $K^{-}p\rightarrow\phi K^{-}p$ scattering experiment found
evidence for a $1^{+}$ state with $M_{0}\sim
1840\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma_{0}\sim
250\mathrm{\,Me\kern-1.00006ptV}$ [50], also seen in the $K^{-}p\rightarrow
K^{-}\pi^{+}\pi^{-}p$ scattering data [55]. Within the large uncertainties the
lower mass state is also consistent with the unconfirmed $K_{1}(1650)$ state
[37], based on evidence from the $K^{-}p\rightarrow\phi K^{-}p$ scattering
experiment [51].

Figure 16: Distributions of (top left) $\phi K^{+}$, (top right)
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ and (bottom)
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ invariant masses for the
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
data (black data points) compared with the results of the default amplitude
fit containing $K^{*+}\rightarrow\phi K^{+}$ and
$X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
contributions. The total fit is given by the red points with error bars.
Individual fit components are also shown. Displays of
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$ and of
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K}$ masses in slices of
$m_{\phi K}$ are shown in Fig. 20.

There is also a substantial $2^{-}$ contribution to the amplitude model. When
modeled as a single resonance ($5.0\sigma$ significant), $M_{0}=1889\pm
27\mathrm{\,Me\kern-1.00006ptV}$ and $\Gamma_{0}=376\pm
94\mathrm{\,Me\kern-1.00006ptV}$ are obtained in agreement with the evidence
from the $K^{-}p\rightarrow\phi K^{-}p$ scattering data which yielded a mass
of around $1840\mathrm{\,Me\kern-1.00006ptV}$ and a width of order
$250\mathrm{\,Me\kern-1.00006ptV}$ [50]. The $K^{+}p\rightarrow\phi K^{+}p$
scattering data also supported such a state at $1810\pm
20\mathrm{\,Me\kern-1.00006ptV}$, but with a narrower width, $140\pm
40\mathrm{\,Me\kern-1.00006ptV}$ [51]. Since two closely spaced $2^{-}$ states
are established from other decay modes [37], and since two $1{\rm D}_{2}$
states are predicted, we allow two resonances in the default fit. The
statistical significance of the second state is $3\sigma$. The masses and
widths obtained by the fit to our data are in good agreement with the
parameters of the $K_{2}(1770)$ and $K_{2}(1820)$ states and in agreement with
the predicted masses of the $1{\rm D}_{2}$ states (Table 3). The individual
fit fractions are poorly defined, and not quoted, because of large destructive
interferences. There is no evidence for an additional $2^{-}$ state in our
data (which could be the expected $2{\rm D}_{2}$ state [53]), but we consider
the inclusion of such a state among the systematic variations.

The most significant $K^{*+}$ resonance in our data is a vector state
($8.5\sigma$). Its mass and width are in very good agreement with the well-
established $K^{*}(1680)$ state, which is observed here in the $\phi K$ decay
mode for the first time, and fits the $1{}^{3}{\rm D}_{1}$ interpretation.
When allowing an extra $1^{-}$ state (candidate for $3{}^{3}{\rm S}_{1}$), its
significance is $2.6\sigma$ with a mass of $1853\pm
5\mathrm{\,Me\kern-1.00006ptV}$, but with a width of only $33\pm
11\mathrm{\,Me\kern-1.00006ptV}$, which cannot be accommodated in the
$\bar{s}u$ quark model. When limiting the width to be
$100\mathrm{\,Me\kern-1.00006ptV}$ or more, the significance drops to
$1.4\sigma$. We do not include it in the default model, but consider its
inclusion as a systematic variation. We also include among the considered
variations the effect of an insignificant ($<0.3\sigma$) tail from the
$K^{*}(1410)$ resonance.

There is a significant ($5.4\sigma$) $2^{+}$ contribution, which we describe
with one very broad resonance, consistent with the claims of a
$K^{*}_{2}(1980)$ state seen in other decays and also consistent with a broad
enhancement seen in $K^{-}p\rightarrow\phi\overline{K}^{0}n$ scattering data
[52]. It can be interpreted as the $2{}^{3}{\rm P}_{2}$ state predicted in
this mass range. An extra $2^{+}$ state added to the model, as suggested e.g.
by the possibility that the $1{}^{3}{\rm F}_{2}$ state is in the probed mass
range, is less than $0.7\sigma$ significant and is considered among the
systematic variations.

There is also $3.5\sigma$ evidence for a $0^{-}$ contribution, consistent with
the previously unconfirmed $K(1830)$ state seen in $K^{-}p\rightarrow\phi
K^{-}p$ scattering data [50]. It could be a $3{}^{1}{\rm S}_{0}$ state. An
extra $0^{-}$ state added to the model (e.g. $4{}^{1}{\rm S}_{0}$) is less
than $0.2\sigma$ significant and is considered among the systematic
variations.

We consider among the systematic variations the inclusion of several further
states that are found not to be significant in the fit. These are a $3^{+}$
state (e.g. $1{\rm F}_{3}$, $<1.8\sigma$), a $4^{+}$ state (e.g. $1{}^{3}{\rm
F}_{4}$, $<2.0\sigma$ or $<0.6\sigma$ if fixed to the $K^{*}_{4}(2045)$
parameters [37]) or a $3^{-}$ state (e.g. $1{}^{3}{\rm D}_{3}$, $<2.0\sigma$
if fixed to the $K^{*}_{3}(1780)$ parameters).

Overall, the $K^{*+}$ composition of our data is in good agreement with the
expectations for the $\bar{s}u$ states, and also in agreement with previous
experimental results on $K^{*}$ states in this mass range. These results add
significantly to the knowledge of $K^{*}$ spectroscopy.

A near-threshold ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
structure in our data is the most significant ($8.4\sigma$) exotic
contribution to our model. We determine its quantum numbers to be
$J^{PC}=1^{++}$ at $5.7\sigma$ significance (Appendix C). When fitted as a
resonance, its mass ($4146.5\pm
4.5\,^{+4.6}_{-2.8}\mathrm{\,Me\kern-1.00006ptV}$) is in excellent agreement
with previous measurements for the $X(4140)$ state, although the width ($83\pm
21\,^{+21}_{-14}\mathrm{\,Me\kern-1.00006ptV}$) is substantially larger. The
upper limit which we previously set for production of a narrow
($\Gamma=15.3\mathrm{\,Me\kern-1.00006ptV}$) $X(4140)$ state based on a small
subset of our present data [21] does not apply to such a broad resonance, i.e.
the present results are consistent with our previous analysis. The statistical
power of the present data sample is not sufficient to study its phase motion
[56]. A model-dependent study discussed in Appendix D suggests that the
$X(4140)$ structure may be affected by the nearby $D_{s}^{\pm}D_{s}^{*\mp}$
coupled-channel threshold. However, larger data samples will be required to
resolve this issue.

We establish the existence of the $X(4274)$ structure with statistical
significance of $6.0\sigma$, at a mass of $4273.3\pm
8.3\,^{+17.2}_{-\phantom{8}3.6}\mathrm{\,Me\kern-1.00006ptV}$ and a width of
$56.2\pm 10.9\,^{+\phantom{8}8.4}_{-11.1}\mathrm{\,Me\kern-1.00006ptV}$. Its
quantum numbers are also $1^{++}$ at $5.8\sigma$ significance. Due to
interference effects, the data peak above the pole mass, underlining the
importance of proper amplitude analysis.

The high $J/\psi\phi$ mass region also shows structures that cannot be
described in a model containing only $K^{*+}$ states. These features are best
described in our model by two $J^{PC}=0^{++}$ resonances at $4506\pm
11\,^{+12}_{-15}$ $\mathrm{\,Me\kern-1.00006ptV}$ and $4704\pm
10\,^{+14}_{-24}\mathrm{\,Me\kern-1.00006ptV}$, with widths of $92\pm
21\,^{+21}_{-20}\mathrm{\,Me\kern-1.00006ptV}$ and $120\pm
31\,^{+42}_{-33}\mathrm{\,Me\kern-1.00006ptV}$, and significances of
$6.1\sigma$ and $5.6\sigma$, respectively. The resonances interfere with a
nonresonant $0^{++}$ ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
contribution that is also significant ($6.4\sigma$). The significances of the
quantum number determinations for the high mass states are $4.0\sigma$ and
$4.5\sigma$, respectively.

Additional $X$ resonances of any $J^{P}$ value ($J\leq 2$) added to our model
have less than $2\sigma$ significance. A modest improvement in fit quality can
be achieved by adding $Z^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}$ resonances to our model, however the significance of such
contributions is too small to justify introducing an exotic hadron
contribution (at most $3.1\sigma$ without accounting for systematic
uncertainty). The parameters obtained for the default model components stay
within their systematic uncertainties when such extra $X$ or $Z^{+}$
contributions are introduced.

Figure 17: Distributions of the fitted decay angles from the $K^{*+}$ decay
chain together with the display of the default fit model described in the
text.  Figure 18: Distributions of the fitted decay angles from the $X$ decay
chain together with the display of the default fit model described in the
text.  Figure 19: Distributions of the fitted decay angles from the $Z$ decay
chain together with the display of the default fit model described in the
text.  Figure 20: Distribution of (left)
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$ and (right)
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K}$ in three slices of
$m_{\phi{K}}$: $<1750\mathrm{\,Me\kern-0.90005ptV}$,
$1750-1950\mathrm{\,Me\kern-0.90005ptV}$, and
$>1950\mathrm{\,Me\kern-0.90005ptV}$ from the top to the bottom, together with
the projections of the default amplitude model. See the legend in Fig. 16 for
a description of the components.

##  9 Summary

In summary, we have performed the first amplitude analysis of
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decays. We have obtained a good description of data in the 6D phase space
including invariant masses and decay angles.

Even though no peaking structures are observed in the $\phi K^{+}$ mass
distributions, correlations in the decay angles reveal a rich spectrum of
$K^{*+}$ resonances. In addition to the angular information contained in the
$K^{*+}$ and $\phi$ decays, the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ decay also helps to probe these resonances, as the helicity states of
the $K^{*+}$ and ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mesons
coming from the $B^{+}$ decay must be equal. Unlike the earlier scattering
experiments investigating $K^{*}\rightarrow\phi K$ decays, we have good
sensitivity to states with both natural and unnatural $J^{P}$ combinations.

The dominant $1^{+}$ partial wave (${\rm FF}=42\pm 8\,^{+5}_{-9}\%$) has a
substantial nonresonant component, and at least one resonance that is
$7.6\sigma$ significant. There is also $2\sigma$ evidence that this structure
can be better described with two resonances matching the expectations for the
two $2{\rm P}_{1}$ excitations of the kaon.

Also prominent is the $2^{-}$ partial wave (${\rm FF}=10.8\pm
2.8\,^{+1.5}_{-4.6}\%$). It contains at least one resonance at $5.0\sigma$
significance. This structure is also better described with two resonances at
$3.0\sigma$ significance. Their masses and widths are in good agreement with
the well established $K_{2}(1770)$ and $K_{2}(1820)$ states, matching the
predictions for the two $1{\rm D}_{2}$ kaon excitations.

The $1^{-}$ partial wave (${\rm FF}=6.7\pm 1.9\,^{+3.2}_{-3.9}\%$) exhibits
$8.5\sigma$ evidence for a resonance which matches the $K^{*}(1680)$ state,
which was well established in other decay modes, and matches the expectations
for the $1{}^{3}{\rm D}_{1}$ kaon excitation. This is the first observation of
its decay to the $\phi K$ final state.

The $2^{+}$ partial wave has a smaller intensity (${\rm FF}=2.9\pm
0.8\,^{+1.7}_{-0.7}\%$), but provides $5.4\sigma$ evidence for a broad
structure that is consistent with the $K_{2}^{*}(1980)$ state observed
previously in other decay modes and matching the expectations for the
$2{}^{3}{\rm P}_{2}$ state.

We also confirm the $K(1830)$ state ($3{}^{1}{\rm S}_{0}$ candidate) at
$3.5\sigma$ significance (${\rm FF}=2.6\pm 1.1\,^{+2.3}_{-1.8}\%$), earlier
observed in the $\phi K$ decay by the $K^{-}p$ scattering experiment. We
determine its mass and width with properly evaluated uncertainties for the
first time.

Overall, our $K^{*+}\rightarrow\phi K^{+}$ results show excellent consistency
with the states observed in other experiments, often in other decay modes, and
fit the mass spectrum predicted for the kaon excitations by the Godfrey-Isgur
model. Most of the $K^{*+}$ structures we observe were previously observed or
hinted at by the $Kp\rightarrow\phi K(p\,\,{\rm or}\,\,n)$ experiments, which
were, however, sometimes inconsistent with each other.

Our data cannot be well described without several
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ contributions. The
significance of the near-threshold $X(4140)$ structure is $8.4\sigma$ with
FF$=13.0\pm 3.2^{+4.8}_{-2.0}\%$. Its width is substantially larger than
previously determined. We determine the $J^{PC}$ quantum numbers of this
structure to be $1^{++}$ at $5.7\sigma$. This has a large impact on its
possible interpretations, in particular ruling out the $0^{++}$ or $2^{++}$
$D_{s}^{*+}D_{s}^{*-}$ molecular models [3, 4, 5, 6, 7, 10]. The
below-${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$-threshold
$D_{s}^{\pm}D_{s}^{*\mp}$ cusp [20, 11] may have an impact on the $X(4140)$
structure, but more data will be required to address this issue. The existence
of the $X(4274)$ structure is established ($6\sigma$) with FF$=7.1\pm
2.5^{+3.5}_{-2.4}\%$ and its quantum numbers are determined to be $1^{++}$
($5.8\sigma$). Together, these two $J^{PC}=1^{++}$ contributions have a fit
fraction of $16\pm 3\,^{+6}_{-2}\%$. Molecular bound-states or cusps cannot
account for the $X(4274)$ $J^{PC}$ values. A hybrid charmonium state would
have $1^{-+}$ [17, 18]. Some tetraquark models expected $0^{-+}$, $1^{-+}$
[13] or $0^{++}$, $2^{++}$ [14] state(s) in this mass range. A tetraquark
model implemented by Stancu [12] not only correctly assigned $1^{++}$ to
$X(4140)$, but also predicted a second $1^{++}$ state at mass not much higher
than the $X(4274)$ mass. Calculations by Anisovich et al. [15] based on the
diquark tetraquark model predicted only one $1^{++}$ state at a somewhat
higher mass. Lebed–Polosa [16] predicted the $X(4140)$ peak to be a $1^{++}$
tetraquark, although they expected the $X(4274)$ peak to be a $0^{-+}$ state
in the same model. A lattice QCD calculation with diquark operators found no
evidence for a $1^{++}$ tetraquark below $4.2\mathrm{\,Ge\kern-1.00006ptV}$
[57].

The high ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ mass region is
investigated with good sensitivity for the first time and shows very
significant structures, which can be described as $0^{++}$ contributions
(${\rm FF}=28\pm 5\pm 7\%$) with a nonresonant term plus two new resonances:
$X(4500)$ ($6.1\sigma$ significant) and $X(4700)$ ($5.6\sigma$). The quantum
numbers of these states are determined with significances of more than
$4\sigma$. The work of Wang et al. [9] predicted a virtual
$D_{s}^{*+}D_{s}^{*-}$ state at $4.48\pm 0.17\mathrm{\,Ge\kern-1.00006ptV}$.
None of the observed ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
states is consistent with the state seen in two-photon collisions by the Belle
collaboration [32].

## Acknowledgements

We thank Eric Swanson for discussions related to his cusp model. We express
our gratitude to our colleagues in the CERN accelerator departments for the
excellent performance of the LHC. We thank the technical and administrative
staff at the LHCb institutes. We acknowledge support from CERN and from the
national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China);
CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); FOM and NWO
(The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FASO
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); NSF (USA). We acknowledge the computing resources that are
provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF
(The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC
(Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany), EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-
Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex
LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal
Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust
(United Kingdom).

##  Appendix A Calculation of decay angles

The decay angles are calculated in a way analogous to that documented in
Appendix IX of Ref. [43]. The five angles for each decay chain are: three
helicity angles of ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, $\phi$
and of the resonance in question (e.g. ${{K}^{*}}$) and two angles between the
decay plane of the resonance and the decay plane of either
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ or $\phi$. In addition, a
rotation is needed to align the muon helicity frames of the $X$ and $Z^{+}$
decay chains to that of the ${{K}^{*}}$ in order to properly describe the
interferences. The choice of ${{K}^{*}}$ as the reference decay chain is
arbitrary. The cosine of a helicity angle of particle $P$, produced in two-
body decay $A\rightarrow P\,B$, and decaying to two particles $P\rightarrow
C\,D$ is calculated from (Eq. (16) in Ref.[43])

$\cos\theta_{P}=-\frac{\vec{p}_{B}\cdot\vec{p}_{C}}{|\vec{p}_{B}|\,|\vec{p}_{C}|},$
(28)

where the momentum vectors are in the rest frame of the particle $P$.

For the $B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*+}$
decay, the angle between the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow\mu^{+}\mu^{-}$ and the $K^{*+}\rightarrow\phi K^{+}$ decay
planes is calculated from666The function atan2$(x,y)$ is the $\tan^{-1}(y/x)$
function with two arguments. The purpose of using two arguments instead of one
is to gather information on the signs of the inputs in order to return the
appropriate quadrant of the computed angle. (Eqs. (14)–(15) in Ref.[43])

$\displaystyle\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$ $\displaystyle={\rm
atan2}(\sin\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\,,\,\cos\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}})$ (29)
$\displaystyle\cos\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$
$\displaystyle=\frac{\vec{a}_{K^{+}}\cdot\vec{a}_{\mu^{+}}}{|\vec{a}_{K^{+}}|\,|\vec{a}_{\mu^{+}}|}$
(30)
$\displaystyle\sin\Delta{\phi}_{{{K}^{*}},{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$
$\displaystyle=\frac{[\vec{p}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\times\vec{a}_{K^{+}}]\cdot\vec{a}_{\mu^{+}}}{|\vec{p}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}|\,|\vec{a}_{K^{+}}|\,|\vec{a}_{\mu^{+}}|}$ (31)
$\displaystyle\vec{a}_{K^{+}}$
$\displaystyle=\vec{p}_{K^{+}}-\frac{\vec{p}_{K^{+}}\cdot\vec{p}_{K^{*+}}}{|\vec{p}_{K^{*+}}|^{2}}\,\vec{p}_{K^{*+}}$
(32) $\displaystyle\vec{a}_{\mu^{+}}$
$\displaystyle=\vec{p}_{\mu^{+}}-\frac{\vec{p}_{\mu^{+}}\cdot\vec{p}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}{|\vec{p}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}|^{2}}\,\vec{p}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}},$
(33)

with all vectors being in the $B^{+}$ rest frame. For the $B^{+}\rightarrow
Z^{+}\phi$ decay, the angle between the
$Z^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ and the
$\phi\rightarrow K^{+}K^{-}$ decay planes, ${\Delta\phi}_{Z,\phi}$, can be
calculated in the same way with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow\phi$, $\mu^{+}\rightarrow K^{+}$ (the $K^{+}$ from the
$\phi$ decay) and the accompanying $K^{+}$ staying the same.

The angle between the decay planes of two sequential decays, e.g. between the
$Z^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ and
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$
decay planes after the $B^{+}\rightarrow Z^{+}\phi$ decay, is calculated from
(Eqs. (18)–(19) in Ref.[43])

$\displaystyle{\Delta\phi}_{Z,{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$
$\displaystyle={\rm
atan2}(\sin{\Delta\phi}_{Z,{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\,,\,\cos{\Delta\phi}_{Z,{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}})$ (34)
$\displaystyle\cos{\Delta\phi}_{Z,{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$
$\displaystyle=\frac{\vec{a}_{\phi}\cdot\vec{a}_{\mu^{+}}}{|\vec{a}_{\phi}|\,|\vec{a}_{\mu^{+}}|}$
(35) $\displaystyle\sin{\Delta\phi}_{Z,{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$
$\displaystyle=\frac{-[\vec{p}_{K^{+}}\times\vec{a}_{\phi}]\cdot\vec{a}_{\mu^{+}}}{|\vec{p}_{K^{+}}|\,|\vec{a}_{\phi}|\,|\vec{a}_{\mu^{+}}|}$
(36) $\displaystyle\vec{a}_{\phi}$
$\displaystyle=\vec{p}_{\phi}-\frac{\vec{p}_{\phi}\cdot\vec{p}_{K^{+}}}{|\vec{p}_{K^{+}}|^{2}}\,\vec{p}_{K^{+}}$
(37) $\displaystyle\vec{a}_{\mu^{+}}$
$\displaystyle=\vec{p}_{\mu^{+}}-\frac{\vec{p}_{\mu^{+}}\cdot\vec{p}_{K^{+}}}{|\vec{p}_{K^{+}}|^{2}}\,\vec{p}_{K^{+}},$
(38)

with all vectors being in the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
rest frame. The other angles of this type are calculated in the same way, with
appropriate substitutions. For example, $\Delta{\phi}_{{{K}^{*}},\phi}$
between the $K^{*+}\rightarrow\phi K^{+}$ and $\phi\rightarrow K^{+}K^{-}$
decay planes after $B^{+}\rightarrow
K^{*+}{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ decay, is calculated
substituting $\phi\longrightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$, $\mu^{+}\longrightarrow K^{+}$ ($K^{+}$ from the $\phi$ decay), and
the accompanying $K^{+}$ staying the same (all vectors are in the $\phi$ rest
frame here).

The angle aligning the muon helicity frames between the $K^{*+}$ and $Z^{+}$
decay chains is calculated from (Eqs. (20)–(21) in Ref.[43])

$\displaystyle\alpha^{Z}$ $\displaystyle={\rm
atan2}(\sin\alpha^{Z}\,,\,\cos\alpha^{Z})$ (39) $\displaystyle\cos\alpha^{Z}$
$\displaystyle=\frac{\vec{a}_{K^{+}}\cdot\vec{a}_{K^{*+}}}{|\vec{a}_{K^{+}}|\,|\vec{a}_{K^{*+}}|}$
(40) $\displaystyle\sin\alpha^{Z}$
$\displaystyle=\frac{-[\vec{p}_{\mu^{+}}\times\vec{a}_{K^{+}}]\cdot\vec{a}_{K^{*+}}}{|\vec{p}_{\mu^{+}}|\,|\vec{a}_{K^{+}}|\,|\vec{a}_{K^{*+}}|}$
(41) $\displaystyle\vec{a}_{K^{*+}}$
$\displaystyle=\vec{p}_{K^{*+}}-\frac{\vec{p}_{K^{*+}}\cdot\vec{p}_{\mu^{+}}}{|\vec{p}_{\mu^{+}}|^{2}}\,\vec{p}_{\mu^{+}}$
(42) $\displaystyle\vec{a}_{K^{+}}$
$\displaystyle=\vec{p}_{K^{+}}-\frac{\vec{p}_{K^{+}}\cdot\vec{p}_{\mu^{+}}}{|\vec{p}_{\mu^{+}}|^{2}}\,\vec{p}_{\mu^{+}},$
(43)

where the $K^{+}$ is the accompanying kaon and all vectors are in the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ rest frame. Similarly,
$\alpha^{X}$ is obtained from the above equations with the
$K^{+}\longrightarrow\phi$ substitution.

For the charge-conjugate
$B^{-}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{-}$
decays, the same formulae apply with the accompanying kaon being $K^{-}$,
$\mu^{+}$ replaced by $\mu^{-}$ and $K^{+}$ from the $\phi$ decay replaced by
the $K^{-}$ from the $\phi$ decay. All azimuthal angles ($\Delta\phi$ and
$\alpha$) have their signs flipped after applying the formulae above (see the
bottom of Appendix IX in Ref. [43]).

##  Appendix B Systematic uncertainty

Individual systematic uncertainties on masses, widths and fit fractions are
presented for $K^{*+}$ contributions in Tables 4–5, and for $X$ contributions
in Table 6. Positive and negative deviations are summed in quadrature
separately for total systematic uncertainties. The statistical uncertainties
are included for comparison.

In many instances, the uncertainty in the $K^{*+}$ model composition is the
dominant systematic uncertainty. The $K^{*+}$ model variations include adding
the following contributions (one-by-one) to the default amplitude model:
second $0^{-}$, $1^{-}$ or $2^{+}$ states, a third $2^{-}$ state, the $3^{-}$
$K_{3}^{*}(1780)$ state, a $3^{+}$ state, the $4^{+}$ $K_{4}^{*}(2045)$ state,
and the below threshold $1^{-}$ $K^{*}(1410)$ state. The variations also
include omitting the second $1^{+}$ or $2^{-}$ states. The observed deviations
in the fit parameters are added in quadrature and then listed in Tables 4–6.

The other sizable source of systematic uncertainty is due to the $L_{B}$ and
$L_{K^{*}}$ (or $L_{X}$) dependence of the Breit–Wigner amplitude in the
numerator of Eq. (9) via Blatt-Weisskopf factors. Helicity states correspond
to mixtures of allowed $L$ values, but we assume the lowest $L$ values in Eq.
(9) in the default fit. We increase $L_{B}$ values by $1$ for all the
components (one-by-one). Values of $L_{K^{*}}$ or $L_{X}$ can only differ by
an even number because of parity conservation in strong decays. We performed
such variations for states in which the higher value is allowed, except for
the $X$ states, since the fit results indicate that the higher $L_{X}$
amplitudes are insignificant. Again, the observed deviations in the fit
parameters are added in quadrature and then listed in Tables 4–6.

The energy release in the
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
decay is small ($\sim 13\%$ on $M_{B}$), and the phase space is very limited,
not offering much range for nonresonant interactions to change. In the default
model the nonresonant terms are represented by constant amplitudes. When
allowing them to change exponentially with mass-squared,
$\exp(-\alpha\,m^{2})$, the slope parameters, $\alpha$, are consistent with
zero. The observed deviations in the measured parameters are included among
the systematic contributions.

Replacing the Breit–Wigner amplitude for the $X(4140)$ structure with a
$D_{s}^{\pm}D_{s}^{*\mp}$ cusp in one particular model (see Appendix D) is
included among the systematic model variations.

The dependence on mass of the total resonance width (Eq. (11)) used in the
default fit assumes that it is dominated by the observed decay mode. All
$K^{*+}$ states are expected to have sizable widths to the other decay modes,
$K\pi$, $K\rho$, $K^{*}(892)\pi$ etc. However, ratios of these partial widths
to the $\phi K$ partial width are unknown. To check the related systematic
uncertainty, we perform an alternative fit (marked $\Gamma_{\rm tot}$ in the
tables) in which the mass dependence of the width is set by the lightest
possible decay mode allowed: $K\pi$ for natural spin-parity resonances and
$K\omega$ for the others. This includes changing the $L_{K^{*}}$ value.

The Blatt-Weisskopf factors contain the $d$ parameter for the effective hadron
size [58] (Eq. (12)), which we set to $3.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}$
in the default fit. As a systematic variation we change its value between
$1.5$ and $5.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}$.

To address the systematic uncertainty in the background parameterization, we
perform amplitude fits with either the left or right $B^{+}$ mass sideband
only. The default fit uses both.

We perform fits to $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K}$
with alternative signal and background parameterizations to determine the
systematic uncertainty on the background fraction in the signal region
($\beta)$. The largest deviation in its value ($\Delta\beta/\beta=+25\%$) is
then used in the alternative amplitude fit to the data.

In the default fit, the simulated events used for the efficiency corrections
are weighted to improve the agreement between the data and the simulation. The
total Monte Carlo event weight ($w^{MC}$) is a product of weights determined
as the ratios between the signal distributions in the data and in the
simulated sample (generated according to the preliminary amplitude model) as
functions of $\mbox{$p_{\mathrm{T}}$}(B^{+})$, number of charged tracks in the
event, and each kaon momentum. These weights are intended to correct for any
inaccuracies in simulation of $pp$ collisions, of $B^{+}$ production
kinematics and in kaon identification. To account for the uncertainty
associated with the efficiency modelling we include among the systematic
variations a fit in which the weights are not applied.

To check the uncertainty related to non-$\phi$ background, we reduce its
fraction by narrowing the $\phi\rightarrow K^{+}K^{-}$ mass selection window
by a factor of two. This also accounts for any uncertainty related to
averaging over this mass in the amplitude fit.

As a cross-check on both the background subtraction and the efficiency
corrections the minimal value of $p_{\mathrm{T}}$ for kaon candidates is
changed from $0.25\mathrm{\,Ge\kern-1.00006ptV}$ to
$0.5\mathrm{\,Ge\kern-1.00006ptV}$, which reduces the background fraction by
54% ($\beta=10.4\%$) and the signal efficiency by 20%, as illustrated in Fig.
21. The mass projections of the fit are shown in Fig. 22. The fit results are
within the assigned total uncertainties as shown at the bottom rows of Tables
4–6.

More details on the systematic error evaluations can be found in Ref. [59].

Figure 21: Mass of $B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi K^{+}$ candidates in the data with the
$\mbox{$p_{\mathrm{T}}$}(K)>250\mathrm{\,Me\kern-0.90005ptV}$ (default) and
$\mbox{$p_{\mathrm{T}}$}(K)>500\mathrm{\,Me\kern-0.90005ptV}$ selection
requirements.

Figure 22: Distributions of (top) $\phi K^{+}$, (middle)
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ and (bottom)
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ invariant masses for the
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
data after changing the
$\mbox{$p_{\mathrm{T}}$}(K)>0.25\mathrm{\,Ge\kern-0.90005ptV}$ requirement to
$\mbox{$p_{\mathrm{T}}$}(K)>0.5\mathrm{\,Ge\kern-0.90005ptV}$, together with
the fit projections. Compare to Fig. 16.  Table 4: Summary of the systematic
uncertainties on the parameters of the $K^{*+}\rightarrow\phi K^{+}$ states
with $J^{P}=2^{-}$ and $1^{+}$. The kaon $p_{\mathrm{T}}$ cross-check results
are given at the bottom. All numbers for masses and widths are in
$\mathrm{\,Me\kern-0.90005ptV}$ and fit fractions in %.

sys. | $2^{-}$ | $K(2^{-})$ | $K^{\prime}(2^{-})$ | $1^{+}$ | $K(1^{+})$ | $K^{\prime}(1^{+})$ | NR
---|---|---|---|---|---|---|---
var.  |  FF  |  $M_{0}$ |  $\Gamma_{0}$ |  FF  |  $M_{0}$ |  $\Gamma_{0}$ |  FF  |  FF  |  $M_{0}$ |  $\Gamma_{0}$ |  FF  |  $M_{0}$ |  $\Gamma_{0}$ |  FF  |  FF
$K^{*}$ |  +1.2  |  +118.1  |  +194.8  |  +4.0  |  +16.2  |  +53.8  |  +4.4  |  +3.9  |  +150.8  |  +122.4  |  +15.6  |  +49.0  |  +159.5  |  +28.5  |  +34.4
model  |  $-$4.1  |  $-$22.3  |  $-$71.0  |  $-$8.6  |  $-$14.9  |  $-$38.5  |  $-$5.5  |  $-$7.5  |  $-$79.2  |  $-$196.2  |  $-$6.1  |  $-$53.8  |  $-$143.2  |  $-$27.2  |  $-$5.1
$L$ |  +0.7  |  +8.6  |  +54.1  |  +3.7  |  +5.5  |  +14.0  |  +3.5  |  +0.8  |  +22.3  |  +20.4  |  +3.4  |  +47.5  |  +37.7  |  +4.8  |  +5.0
var.  |  $-$1.5  |  $-$63.3  |  $-$127.9  |  $-$9.3  |  $-$31.2  |  $-$59.5  |  $-$8.6  |  $-$2.2  |  $-$48.6  |  $-$70.3  |  $-$0.9  |  $-$159.9  |  $-$72.5  |  $-$8.7  |  $-$2.2
NR exp.  |  +0.5  |  $-$4.8  |  $-$13.5  |  +0.4  |  $-$0.6  |  +8.6  |  +1.8  |  $-$2.2  |  $-$5.9  |  $-$3.7  |  +0.7  |  $-$21.4  |  $-$45.4  |  +0.8  |  +0.3
$X$ cusp  |  +0.0  |  +24.6  |  +42.2  |  +5.4  |  $-$0.8  |  +10.8  |  +3.8  |  +1.8  |  +4.5  |  +5.5  |  +4.4  |  $-$12.0  |  +40.6  |  +8.4  |  $-$0.3
$\Gamma_{\rm tot}$ |  $-$0.2  |  +0.8  |  +38.7  |  $-$1.6  |  $-$1.9  |  $-$12.6  |  $-$2.4  |  +0.6  |  $-$29.5  |  +17.2  |  +0.9  |  $-$0.1  |  +7.1  |  $-$2.3  |  +2.2
d=1.5  |  +0.1  |  +18.2  |  +67.2  |  $-$0.6  |  +2.7  |  +6.0  |  $-$1.5  |  +0.7  |  $-$17.4  |  $-$5.6  |  $-$1.0  |  +8.2  |  +13.9  |  $-$2.1  |  +1.7
d=5.0  |  +0.2  |  $-$7.2  |  $-$25.8  |  $-$0.1  |  $-$1.0  |  $-$0.5  |  +1.3  |  $-$1.5  |  +12.2  |  $-$6.9  |  +0.5  |  $-$8.4  |  $-$42.8  |  $-$1.0  |  $-$1.5
Left s.  |  +0.1  |  $-$4.2  |  $-$9.5  |  $-$0.2  |  $-$1.1  |  +2.0  |  +0.9  |  $-$1.0  |  +0.9  |  +0.2  |  +1.1  |  $-$8.7  |  $-$30.2  |  +0.9  |  +0.3
Right s.  |  $-$0.1  |  +3.2  |  +5.0  |  $-$0.4  |  +3.8  |  +0.1  |  $-$1.2  |  +1.2  |  $-$1.3  |  +12.5  |  $-$0.4  |  +11.6  |  +36.5  |  $-$1.5  |  $-$0.1
$\beta$ |  +0.2  |  $-$8.1  |  $-$35.4  |  +1.7  |  $-$9.3  |  $-$6.7  |  +2.6  |  $-$2.7  |  +28.0  |  $-$8.2  |  +4.0  |  $-$23.4  |  $-$63.0  |  +4.8  |  $-$0.8
No $w^{MC}$ |  $-$0.8  |  $-$0.2  |  +0.4  |  $-$1.1  |  +0.0  |  $-$0.5  |  $-$1.5  |  $-$0.8  |  +1.9  |  +1.2  |  +0.1  |  +0.6  |  +1.8  |  $-$0.7  |  +0.7
$\phi$ window  |  $-$1.0  |  $-$25.0  |  $-$27.2  |  $-$2.6  |  $-$1.1  |  +41.2  |  $-$1.4  |  $-$2.7  |  $-$11.3  |  $-$36.5  |  +0.0  |  $-$15.2  |  $-$23.1  |  +6.0  |  $-$1.9
Total  |  +1.5  |  +122.3  |  +220.7  |  +7.7  |  +17.7  |  +82.0  |  +7.2  |  +4.7  |  +153.0  |  +138.0  |  +16.7  |  +69.7  |  +173.5  |  +31.3  |  +34.5
sys.  |  $-$4.6  |  $-$76.5  |  $-$154.3  |  $-$13.3  |  $-$34.7  |  $-$72.0  |  $-$10.9  |  $-$9.2  |  $-$100.5  |  $-$214.8  |  $-$6.3  |  $-$172.3  |  $-$177.9  |  $-$28.8  |  $-$6.4
Stat.  |  -2.8  |  -34.9  |  -116.3  |  -11.0  |  -26.6 |  -58.1 |  -11.2  |  -8.1  |  -59.0  |  -157.0 |  -10.3  |  -65.0  |  -170.3 |  -20.4  |  -13.1
$\mbox{$p_{\mathrm{T}}$}^{K}\\!\\!>$500  |  $-$2.7  |  $-$0.4  |  +4.9  |  $-$3.7  |  $-$10.1  |  $-$67.0  |  $-$5.7  |  +6.4  |  +95.2  |  $-$238.7  |  $-$3.7  |  $-$87.7  |  +33.6  |  $-$3.8  |  +4.7

Table 5: Summary of the systematic uncertainties on the parameters of the $K^{*+}\rightarrow\phi K^{+}$ states with $J^{P}=0^{-}$, $1^{-}$ and $2^{+}$. The kaon $p_{\mathrm{T}}$ cross-check results are given at the bottom. All numbers for masses and widths are in $\mathrm{\,Me\kern-0.90005ptV}$ and fit fractions in %. sys. | $K^{*}(1^{-})$ | $K(0^{-})$ | $K^{*}(2^{+})$
---|---|---|---
var.  |  $M_{0}$ |  $\Gamma_{0}$ |  FF  |  $M_{0}$ |  $\Gamma_{0}$ |  FF  |  $M_{0}$ |  $\Gamma_{0}$ |  FF
${{K}^{*}}$ |  +19.9  |  +31.4  |  +2.6  |  +54.8  |  +236.9  |  +1.7  |  +214.3  |  +805.2  |  +1.6
model  |  $-$33.1  |  $-$141.0  |  $-$2.7  |  $-$90.2  |  $-$96.3  |  $-$1.7  |  $-$66.9  |  $-$223.8  |  $-$0.6
$L$ |  +14.2  |  +59.3  |  +1.8  |  +12.8  |  +51.6  |  +0.7  |  +52.0  |  +172.3  |  +0.3
var.  |  $-$17.7  |  $-$44.7  |  $-$0.2  |  $-$44.4  |  $-$31.1  |  $-$0.2  |  $-$19.1  |  $-$107.4  |  $-$0.3
NR exp.  |  +3.3  |  +11.5  |  +0.2  |  $-$22.9  |  +36.3  |  +0.4  |  $-$13.7  |  $-$65.1  |  +0.0
$X$ cusp  |  +4.5  |  +5.5  |  $-$1.2  |  +7.8  |  +11.4  |  +0.1  |  +26.5  |  +6.1  |  $-$0.2
$\Gamma_{\rm tot}$ |  $-$101.5  |  $-$93.1  |  +0.2  |  $-$2.8  |  $-$6.2  |  $-$0.1  |  $-$167.6  |  $-$230.0  |  +0.3
d=1.5  |  +21.1  |  +121.7  |  +0.0  |  +12.1  |  +2.5  |  $-$0.1  |  +102.2  |  +806.2  |  +0.0
d=5.0  |  $-$4.9  |  $-$21.0  |  +0.0  |  $-$10.3  |  +6.3  |  +0.2  |  $-$72.0  |  $-$242.5  |  +0.0
Left s.  |  +2.7  |  +7.7  |  +0.0  |  $-$12.6  |  20.1  |  +0.2  |  $-$17.9  |  $-$28.8  |  +0.2
Right s.  |  $-$3.0  |  +7.7  |  +0.0  |  +10.0  |  $-$23.5  |  $-$0.2  |  +19.2  |  +24.7  |  $-$0.2
$\beta$ |  +2.2  |  $-$4.1  |  +0.1  |  $-$43.0  |  +32.2  |  +0.5  |  $-$18.5  |  +1.1  |  +0.4
No $w^{MC}$ |  +0.2  |  $-$0.4  |  +0.1  |  +1.0  |  $-$2.4  |  $-$0.4  |  $-$0.4  |  $-$3.1  |  $-$0.2
$\phi$ window  |  +0.5  |  $-$28.9  |  $-$1.8  |  $-$33.6  |  +94.5  |  +0.9  |  $-$97.0  |  $-$258.9  |  +0.2
Total  |  +32.9  |  +139.8  |  +3.2  |  +59.0  |  +280.2  |  +2.3  |  +245.2  |  +1152.7  |  +1.7
sys.  |  $-$108.4  |  $-$180.7  |  $-$3.9  |  $-$114.8  |  $-$104.1  |  $-$1.8  |  $-$239.7  |  $-$559.0  |  $-$0.7
Stat.  |  19.9  |  74.7  |  1.9  |  43.2  |  90.4  |  1.1  |  94.2  |  310.6  |  0.8
$\mbox{$p_{\mathrm{T}}$}^{K}\\!\\!>$500  |  $-$15.6  |  $-$47.1  |  $-$0.2  |  $-$161.9  |  $-$2.4  |  $-$0.2  |  $-$10.1  |  $-$102.2  |  $-$0.1
Table 6: Summary of the systematic uncertainties on the parameters of the
$X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ states. The
kaon $p_{\mathrm{T}}$ cross-check results are given at the bottom. All numbers
for masses and widths are in $\mathrm{\,Me\kern-0.90005ptV}$ and fit fractions
in %.

sys. | $1^{+}$ | $X(4140)$ | $X(4274)$ | $0^{+}$ | $X(4500)$ | $X(4700)$ | NR
---|---|---|---|---|---|---|---
var.  |  FF  |  $M_{0}$ |  $\Gamma_{0}$ |  FF  |  $M_{0}$ |  $\Gamma_{0}$ |  FF  |  FF  |  $M_{0}$ |  $\Gamma_{0}$ |  FF  |  $M_{0}$ |  $\Gamma_{0}$ |  FF  |  FF
${{K}^{*}}$ |  +2.0  |  +3.6  |  +17.1  |  +2.2  |  +11.2  |  +7.9  |  +1.4  |  +1.8  |  +9.3  |  +13.8  |  +2.0  |  +7.5  |  +38.6  |  +6.7  |  +8.0
model  |  $-$1.7  |  $-$2.6  |  $-$11.7  |  $-$1.9  |  $-$2.5  |  $-$8.5  |  $-$1.5  |  $-$11.0  |  $-$8.6  |  $-$16.6  |  $-$1.7  |  $-$18.9  |  $-$13.5  |  $-$4.8  |  $-$16.6
$L$ |  +3.2  |  +2.2  |  +7.3  |  +2.1  |  +10.6  |  +1.4  |  +1.0  |  +0.3  |  +1.3  |  +10.8  |  +1.7  |  +9.0  |  +12.4  |  +1.5  |  +1.2
var.  |  +0.0  |  $-$1.2  |  $-$6.2  |  $-$0.5  |  $-$0.8  |  $-$4.6  |  $-$1.2  |  $-$4.7  |  $-$9.6  |  $-$11.2  |  $-$1.6  |  $-$6.8  |  $-$24.9  |  $-$0.8  |  $-$8.5
NR exp.  |  +0.4  |  $-$0.2  |  $-$0.1  |  +0.4  |  $-$0.2  |  +0.6  |  +0.8  |  $-$1.7  |  +6.3  |  +0.3  |  +0.2  |  +7.1  |  $-$15.7  |  $-$1.7  |  $-$9.1
$X$ cusp  |  +2.2  |  |  |  +0.9  |  +6.4  |  $-$5.4  |  $-$1.4  |  $-$1.2  |  +0.0  |  +1.2  |  +0.2  |  +1.9  |  $-$2.5  |  0.5  |  $-$1.6
$\Gamma_{\rm tot}$ |  $-$0.6  |  +0.2  |  +1.5  |  $-$0.4  |  +3.2  |  +0.2  |  $-$0.3  |  +0.1  |  +0.8  |  $-$0.1  |  $-$0.3  |  +0.9  |  $-$5.8  |  $-$0.9  |  $-$1.1
d=1.5  |  $-$0.9  |  +1.1  |  +5.3  |  $-$0.5  |  +2.2  |  +0.8  |  $-$0.4  |  +0.5  |  +1.7  |  +3.2  |  +0.1  |  $-$0.1  |  +1.7  |  +0.0  |  +1.1
d=5.0  |  +1.1  |  $-$0.2  |  $-$2.0  |  +0.6  |  +0.2  |  $-$0.8  |  +0.3  |  $-$0.5  |  $-$1.0  |  $-$3.1  |  $-$0.1  |  $-$1.2  |  $-$3.2  |  $-$0.7  |  $-$2.5
Left s.  |  +0.1  |  $-$0.4  |  $-$2.0  |  +0.1  |  +0.4  |  $-$0.8  |  +0.1  |  $-$0.5  |  $-$2.4  |  $-$2.6  |  $-$0.2  |  $-$1.5  |  $-$3.1  |  $-$0.7  |  $-$1.2
Right s.  |  $-$0.3  |  +0.3  |  +2.6  |  $-$0.2  |  $-$0.6  |  +1.0  |  +0.0  |  +0.5  |  +3.7  |  +3.4  |  +0.4  |  +1.2  |  +7.0  |  +0.8  |  +1.6
$\beta$ |  +1.2  |  $-$0.6  |  $-$3.6  |  +1.2  |  +1.7  |  $-$0.7  |  +0.9  |  $-$2.5  |  $-$4.6  |  $-$11.1  |  $-$0.5  |  $-$3.9  |  $-$6.1  |  $-$1.4  |  $-$1.4
No $w^{MC}$ |  +1.6  |  +0.0  |  +0.0  |  +0.1  |  +0.0  |  +0.0  |  +1.4  |  +1.7  |  +0.0  |  +0.2  |  +0.2  |  +0.1  |  +0.0  |  +1.2  |  +2.7
$\phi$ window  |  +2.5  |  +1.1  |  +4.7  |  +2.4  |  $-$1.6  |  +1.4  |  +1.8  |  +4.2  |  $-$4.3  |  +7.1  |  +1.2  |  $-$9.3  |  +5.8  |  +0.7  |  +4.7
Total  |  +5.9  |  +4.6  |  +20.7  |  +4.7  |  +17.2  |  +8.4  |  +3.5  |  +6.5  |  +12.0  |  +20.8  |  +3.2  |  +13.9  |  +42.0  |  +7.2  |  +11.0
sys.  |  $-$2.1  |  $-$2.8  |  $-$13.5  |  $-$2.0  |  $-$3.6  |  $-$11.1  |  $-$2.4  |  $-$6.7  |  $-$14.5  |  $-$20.4  |  $-$2.3  |  $-$24.1  |  $-$33.3  |  $-$5.3  |  $-$21.0
Stat.  |  2.8  |  4.5  |  20.7  |  3.2  |  8.3  |  10.9  |  2.5  |  5.1  |  11.1  |  21.2  |  2.4  |  10.1  |  30.7  |  4.9  |  10.7
$\mbox{$p_{\mathrm{T}}$}^{K}\\!\\!>$500  |  $-$1.3  |  +1.6  |  +1.7  |  $-$2.7  |  +7.8  |  +12.2  |  +0.2  |  $-$9.6  |  $-$10.9  |  $-$18.6  |  $-$3.2  |  $-$4.7  |  $-$12.7  |  $-$6.6  |  $-$17.1

##  Appendix C Spin analysis for the
$X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ states

To determine the quantum numbers of each $X$ state, fits are done under
alternative $J^{PC}$ hypotheses. The likelihood-ratio test is used to quantify
rejection of these hypotheses. Since different spin-parity assignments are
represented by different functions in the angular part of the fit PDF, they
represent separate hypotheses. For two models representing separate
hypotheses, assuming a $\chi^{2}$ distribution with one degree of freedom for
$\Delta(\\!-2\\!\ln{\cal L})$ under the disfavored $J^{PC}$ hypothesis gives a
lower limit on the significance of its rejection [60]. The results for the
default fit approach are shown in Table 7. The $J^{PC}$ values of the
$X(4140)$ and $X(4274)$ states are both determined to be $1^{++}$ with
$7.6\sigma$ and $6.4\sigma$ significance, respectively. The quantum numbers of
$X(4500)$ and of $X(4700)$ states are both established to be $0^{++}$ at
$5.2\sigma$ and $4.9\sigma$ level, respectively.

The separation from the alternative $J^{PC}$ hypothesis with likelihood
closest to that for the favored quantum numbers in the default fit is studied
for each state under the fit variations which have dominant effects on the
resonance parameters as shown in Table 8. The lowest values are taken for the
final significances of the quantum number determinations: $5.7\sigma$ for
$X(4140)$, $5.8\sigma$ for $X(4274)$, $4.0\sigma$ for $X(4500)$ and
$4.5\sigma$ for $X(4700)$.

Table 7: Statistical significance of $J^{PC}$ preference for the $X$ states in the default model. The lowest significance value for each state is highlighted.  $J^{PC}$ | $X(4140)$ | $X(4274)$ | $X(4500)$ | $X(4700)$
---|---|---|---|---
$0^{++}$ | 10.3$\sigma$ | 7.8$\sigma$ | preferred | preferred
$0^{-+}$ | 12.5$\sigma$ | 7.0$\sigma$ | 8.1$\sigma$ | 8.2$\sigma$
$1^{++}$ | preferred | preferred | $\mathbf{5.2\sigma}$ | $\mathbf{4.9\sigma}$
$1^{-+}$ | 10.4$\sigma$ | $\mathbf{6.4\sigma}$ | 6.5$\sigma$ | 8.3$\sigma$
$2^{++}$ | $\mathbf{7.6\sigma}$ | 7.2$\sigma$ | 5.6$\sigma$ | 6.8$\sigma$
$2^{-+}$ | 9.6$\sigma$ | $\mathbf{6.4\sigma}$ | 6.5$\sigma$ | 6.3$\sigma$
Table 8: Significance, in standard deviations, of $J^{PC}$ preference for the $X$ states for dominant systematic variations of the fit model. The label “$L+n$” specifies which $L$ value in Eq. (9) is increased relative to its minimal value and by how much ($n$). The lowest significance value for each state is highlighted. systematic variation | $X(4140)$ | $X(4274)$ | $X(4500)$ | $X(4700)$
---|---|---|---|---
alternative $J^{PC}$ | $2^{++}$ | $1^{-+}$ | $2^{-+}$ | $1^{++}$ | $1^{++}$
default fit | 7.6 | 6.4 | 6.4 | 5.2 | 4.9
$K^{\prime}(1^{+})$ $L_{{{K}^{*}}}$ \+ 2 | 12.2 | 6.2 | 7.4 | 5.4 | 5.1
$K_{2}(2^{-})$ $L_{{{K}^{*}}}$ \+ 2 | 5.7 | 6.0 | 5.8 | 5.2 | 4.5
${K^{*}_{3}(1780)}$ included | 6.2 | 6.6 | 6.3 | 4.9 | 4.5
extra $K^{*}(1^{-})$ included | 6.8 | 6.1 | 5.8 | 5.8 | 4.7
extra $K_{2}(2^{-})$ included | 6.9 | 6.7 | 6.2 | 4.0 | 4.8
NR exp | 7.5 | 6.5 | 6.1 | 8.9 | 4.7

##  Appendix D Is $X(4140)$ a $D_{s}^{\pm}D_{s}^{*\mp}$ cusp?

While our $1^{++}$ assignment to $X(4140)$ and its large width rule out an
interpretation as a $0^{++}$ or $2^{++}$ $D_{s}^{*+}D_{s}^{*-}$ molecule (for
which $1^{++}$ is not allowed [3]) with large $\sim
83\mathrm{\,Me\kern-1.00006ptV}$ binding energy as suggested by many authors
[3, 4, 5, 6, 7], such a structure could be formed by molecular forces in a
$D_{s}^{\pm}D_{s}^{*\mp}$ pair in S-wave [20, 11]. Since the sum of
$D_{s}^{\pm}$ and $D_{s}^{*\mp}$ masses ($4080\mathrm{\,Me\kern-1.00006ptV}$)
is below the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ mass
threshold ($4116$ $\mathrm{\,Me\kern-1.00006ptV}$), such a contribution would
not be described by the Breit–Wigner function with a pole above that
threshold. The investigation of all possible parameterizations for such
contributions, which are model dependent, goes beyond the scope of this
analysis. However, we attempt a fit with a simple threshold cusp
parameterization proposed by Swanson (Ref. [61] and private communications),
in which the introduction of an exponential form factor, with a momentum scale
($\beta_{0}$) characterizing the hadron size, makes the cusp peak slightly
above the sum of masses of the rescattering mesons. While controversial [62],
this model provided a successful description of the $Z_{c}(3900)^{+}$ and
$Z_{c}(4025)^{+}$ exotic meson candidates with masses peaking slightly above
the molecular thresholds [61].

In Swanson’s model a virtual loop with two mesons $A$ and $B$ inside (Fig. 1
left in Ref. [61]) contributes, in the non-relativistic near-threshold
approximation, the following amplitude,

$\Pi(m)=\int\,\frac{d^{3}q}{(2\pi)^{3}}\,\frac{q^{2l}\,e^{-2q^{2}/\beta_{0}^{2}}}{m-M_{A}-M_{B}-\frac{q^{2}}{2\,\mu_{AB}}+i\,\epsilon},$
(44)

where $m$ is ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ mass,
$\mu_{AB}=M_{A}\,M_{B}/(M_{A}+M_{B})$ is the reduced mass of the pair,
$\beta_{0}$ is a hadronic scale of order of $\Lambda_{\rm QCD}$, (which can be
$AB$ dependent), $\epsilon$ is a very small number ($\epsilon\rightarrow 0$),
$l$ is the angular momentum between $A$ and $B$. The lowest $l$ values are
expected to dominate. The amplitude $\Pi(m)$ reflects coupled-channel
kinematics. The above integral can be conveniently expressed as

$\displaystyle\Pi(m)$
$\displaystyle=-\frac{\mu_{AB}\,\beta_{0}}{\sqrt{2}\pi^{2}}\,I(Z)$ (45)
$\displaystyle Z$
$\displaystyle=\frac{4\mu_{AB}}{\beta_{0}^{2}}(M_{A}+M_{B}-m)$ (46)
$\displaystyle I(Z)$
$\displaystyle=\int_{0}^{\infty}\,dx\,\frac{x^{2+2l}\,e^{-x^{2}}}{x^{2}+Z-i\,\epsilon},$
(47)

where $-Z$ is the scaled mass deviation from the $AB$ threshold. For $l=0$,
the integral above evaluates to

$I(Z)=\frac{1}{2}\,\sqrt{\pi}[1-\sqrt{\pi\,Z}\,e^{Z}\,{\rm erfc}(\sqrt{Z})].$
(48)

For masses below the $AB$ threshold $Z>0$ and $I(Z)$ (thus $\Pi(Z)$) has no
imaginary part. For masses above the threshold $Z<0$, $\sqrt{Z}$ is imaginary,
which leads to both real and imaginary parts. The real and imaginary parts of
$-I(Z)$ as a function of $-Z$ are shown in Fig. 23, while the corresponding
Argand diagram is shown in Fig. 24 where it is compared to the phase motion of
the Breit–Wigner function.

Figure 23: Dependence of the real and imaginary parts of the cusp amplitude on
the mass in Swanson’s model [61]. See the text for a more precise explanation.
Figure 24: The Argand diagram of the the cusp amplitude in Swanson’s model
[61]. Motion with the mass is counter-clockwise. The peak amplitude is reached
at threshold when the real part is maximal and the imaginary part is zero. The
Breit–Wigner amplitude gives circular phase motion, also with counter-
clockwise mass evolution, with maximum magnitude when zero is crossed on the
real axis.

The function $\Pi(m)$ replaces the Breit–Wigner function
$BW(m|M_{0},\Gamma_{0})$ in Eq. (10). The Blatt–Weisskopf functions in Eq. (9)
still apply. Thus, the functional form of this representation, has three free
parameters to determine from the data ($\beta_{0}$ and the complex S-wave
helicity coupling). The value of $\beta_{0}$ obtained by the fit to the data,
$297\pm 20\mathrm{\,Me\kern-1.00006ptV}$, is close to the value of
$300\mathrm{\,Me\kern-1.00006ptV}$ with which Swanson was successful in
describing the other near-threshold exotic meson candidates [61]. A fit with
such parameterization (see Fig. 25 for mass distributions), has a better
likelihood than the Breit–Wigner fit by $1.6\sigma$ for the default model (8
free parameters in the $X(4140)$ Breit–Wigner parameterization), and better by
$3\sigma$ when only S-wave couplings are allowed (4 free parameters),
providing an indication that the $X(4140)$ structure may not be a bound state
that can be described by the Breit–Wigner formula. Larger data samples will be
required to obtain more insight. We have included the $X(4140)$ cusp model
among the systematic variations considered for parameters of the other fit
components. The differences between the results obtained with the default
amplitude model and the model in which the $X(4140)$ structure is represented
by a cusp are given in Tables 4–6.

Figure 25: Distributions of (top left) $\phi K^{+}$, (top right)
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ and (bottom)
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ invariant masses for the
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
data (black data points) compared with the results of the amplitude fit
containing $K^{*+}\rightarrow\phi K^{+}$ and
$X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ contributions
in which $X(4140)$ is represented as a $J^{PC}=1^{++}$ $D_{s}^{+}D_{s}^{*-}$
cusp. The total fit is given by the red points with error bars. Individual fit
components are also shown.

The $X(4274)$ mass structure can be reasonably well described by the $0^{-+}$
cusp model for $D_{s}^{\pm}D_{s0}^{*}(2317)^{\mp}$ scattering (Fig. 26).
However, the multidimensional likelihood is substantially worse than for the
default amplitude model ($6.6\sigma$). The likelihood remains worse for the
default fit even if $1^{++}$ quantum numbers are assumed for such a cusp
($4.4\sigma$). This particular cusp parameterization is not useful when trying
to describe any of the higher mass ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ structures.

Figure 26: Distributions of ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ invariant mass for the
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi K^{+}$
data (black data points) compared with the results of the amplitude fit
containing $K^{*+}\rightarrow\phi K^{+}$ and
$X\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ contributions
in which $X(4140)$ and $X(4274)$ are represented as $J^{PC}=1^{++}$
$D_{s}^{\pm}D_{s}^{*\mp}$ and $0^{-+}$ $D_{s}^{\pm}D_{s0}^{*}(2317)^{\mp}$
cusps, respectively. The total fit is given by the red points with error bars.
Individual fit components are also shown.

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LHCb collaboration

R. Aaij39, B. Adeva38, M. Adinolfi47, Z. Ajaltouni5, S. Akar6, J. Albrecht10,
F. Alessio39, M. Alexander52, S. Ali42, G. Alkhazov31, P. Alvarez Cartelle54,
A.A. Alves Jr58, S. Amato2, S. Amerio23, Y. Amhis7, L. An40, L. Anderlini18,
G. Andreassi40, M. Andreotti17,g, J.E. Andrews59, R.B. Appleby55, O. Aquines
Gutierrez11, F. Archilli1, P. d’Argent12, J. Arnau Romeu6, A. Artamonov36, M.
Artuso60, E. Aslanides6, G. Auriemma26, M. Baalouch5, I. Babuschkin55, S.
Bachmann12, J.J. Back49, A. Badalov37, C. Baesso61, W. Baldini17, R.J.
Barlow55, C. Barschel39, S. Barsuk7, W. Barter39, V. Batozskaya29, B.
Batsukh60, V. Battista40, A. Bay40, L. Beaucourt4, J. Beddow52, F. Bedeschi24,
I. Bediaga1, L.J. Bel42, V. Bellee40, N. Belloli21,i, K. Belous36, I.
Belyaev32, E. Ben-Haim8, G. Bencivenni19, S. Benson39, J. Benton47, A.
Berezhnoy33, R. Bernet41, A. Bertolin23, F. Betti15, M.-O. Bettler39, M. van
Beuzekom42, I. Bezshyiko41, S. Bifani46, P. Billoir8, T. Bird55, A.
Birnkraut10, A. Bitadze55, A. Bizzeti18,u, T. Blake49, F. Blanc40, J. Blouw11,
S. Blusk60, V. Bocci26, T. Boettcher57, A. Bondar35, N. Bondar31,39, W.
Bonivento16, A. Borgheresi21,i, S. Borghi55, M. Borisyak67, M. Borsato38, F.
Bossu7, M. Boubdir9, T.J.V. Bowcock53, E. Bowen41, C. Bozzi17,39, S. Braun12,
M. Britsch12, T. Britton60, J. Brodzicka55, E. Buchanan47, C. Burr55, A.
Bursche2, J. Buytaert39, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,i, M.
Calvo Gomez37,m, P. Campana19, D. Campora Perez39, L. Capriotti55, A.
Carbone15,e, G. Carboni25,j, R. Cardinale20,h, A. Cardini16, P. Carniti21,i,
L. Carson51, K. Carvalho Akiba2, G. Casse53, L. Cassina21,i, L. Castillo
Garcia40, M. Cattaneo39, Ch. Cauet10, G. Cavallero20, R. Cenci24,t, M.
Charles8, Ph. Charpentier39, G. Chatzikonstantinidis46, M. Chefdeville4, S.
Chen55, S.-F. Cheung56, V. Chobanova38, M. Chrzaszcz41,27, X. Cid Vidal38, G.