EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2022-196 LHCb-PAPER-2022-028 1 November 2022

Study of the ${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
3.73305pt\overline{\kern-3.73305pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
decay

LHCb collaboration

The decay ${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
is studied in proton-proton collisions at a center-of-mass energy of
$\sqrt{s}=13$ TeV using data corresponding to an integrated luminosity of 5
$\text{\,fb}^{-1}$ collected by the LHCb experiment. In the ${{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}$ system, the ${{\mathchar
28932\relax}_{c}}(2930)^{0}$ state observed at the BaBar and Belle experiments
is resolved into two narrower states, ${{\mathchar
28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar 28932\relax}_{c}}(2939)^{0}$,
whose masses and widths are measured to be

$\displaystyle m({{\mathchar 28932\relax}_{c}}(2923)^{0})$
$\displaystyle=2924.5\pm 0.4\pm 1.1\text{\,Me\kern-1.00006ptV},$
$\displaystyle m({{\mathchar 28932\relax}_{c}}(2939)^{0})$
$\displaystyle=2938.5\pm 0.9\pm 2.3\text{\,Me\kern-1.00006ptV},$
$\displaystyle\Gamma({{\mathchar 28932\relax}_{c}}(2923)^{0})$
$\displaystyle=\phantom{000}4.8\pm 0.9\pm 1.5\text{\,Me\kern-1.00006ptV},$
$\displaystyle\Gamma({{\mathchar 28932\relax}_{c}}(2939)^{0})$
$\displaystyle=\phantom{00}11.0\pm 1.9\pm 7.5\text{\,Me\kern-1.00006ptV},$

where the first uncertainties are statistical and the second systematic. The
results are consistent with a previous LHCb measurement using a prompt
${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$ sample. Evidence of a new
${{\mathchar 28932\relax}_{c}}(2880)^{0}$ state is found with a local
significance of $3.8\,\sigma$, whose mass and width are measured to be
$2881.8\pm 3.1\pm 8.5\text{\,Me\kern-1.00006ptV}$ and $12.4\pm 5.3\pm
5.8\text{\,Me\kern-1.00006ptV}$, respectively. In addition, evidence of a new
decay mode ${{\mathchar 28932\relax}_{c}}(2790)^{0}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}$ is found with a significance of $3.7\,\sigma$.
The relative branching fraction of ${{{B}^{-}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$ with respect to the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ decay is measured to be
$2.36\pm 0.11\pm 0.22\pm 0.25$, where the first uncertainty is statistical,
the second systematic and the third originates from the branching fractions of
charm hadron decays.

Submitted to Phys. Rev. D

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

##  1 Introduction

The quark model predicts a rich spectrum of singly charmed baryons. However,
many states, especially those containing at least one $s$ quark (e.g.
${{\mathchar 28932\relax}^{0}_{c}}$, ${{\mathchar 28938\relax}^{0}_{c}}$),
have not been experimentally established. An effective approach to search for
charm baryons is in ${b}$-hadron decays. Upon the observation of the
${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
decay,111The inclusion of charge-conjugate processes is implied throughout the
paper. the BaBar collaboration reported evidence of a ${{\mathchar
28932\relax}_{c}}(2930)^{0}$ state in the ${{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}$ system with a low significance[1]; and later
the Belle collaboration confirmed this with a larger yield[2]. Recently, the
Belle collaboration also found evidence of its charged counterpart in the
$B^{0}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{0}_{\mathrm{S}}}$ decay[3]. The LHCb experiment,
with a much higher $B$ meson yield and excellent momentum resolution, should
be able to confirm the ${{\mathchar 28932\relax}_{c}}(2930)^{0}$ state by
examining the ${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
decay, precisely measure its properties and resolve a potential finer
structure.

Another way to look for charmed baryons is to search for their production in
primary $pp$ interactions, recently exploited by the LHCb experiment [4, 5, 6,
7]. A search for excited ${{\mathchar 28932\relax}^{0}_{c}}$ baryons in prompt
${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$ pairs performed at the LHCb
experiment found that there are two narrow states, ${{\mathchar
28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar 28932\relax}_{c}}(2939)^{0}$, at
the position of ${{\mathchar 28932\relax}_{c}}(2930)^{0}$ observed at $B$
factories, with the presence of an additional ${{\mathchar
28932\relax}_{c}}(2965)^{0}$ structure at higher mass[7]. At lower mass, a
peaking structure is seen at around 2880 MeV,222Natural units with $\hbar=c=1$
are used throughout this paper. noted as ${{\mathchar
28932\relax}_{c}}(2880)^{0}$, which is obscured by partially reconstructed
background but is in the position of an expected resonance [8]. The complexity
of the current experimental situation motivates the study of the
${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
decay at the LHCb experiment. Although the exclusive reactions have lower
yield compared to the prompt search, the background level is expected to be
lower.

Furthermore, the ${{{B}^{-}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$ decay provides a unique opportunity to
search for possible exotic candidates in the ${{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
spectra. The Belle collaboration observed a significant near-threshold
enhancement in the ${{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$ mass
spectrum through the reaction ${e^{+}}{e^{-}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}\gamma_{\mathrm{ISR}}$, where $\gamma_{\mathrm{ISR}}$
indicates a high momentum photon emitted in the initial state radiation
process [9]. The BESIII experiment recently measured the
${e^{+}}{e^{-}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$ cross-
section with high precision, and an enhanced value near threshold was reported
[10]. If the enhancement of the ${{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$
production around $4600\text{\,Me\kern-1.00006ptV}$ is due to the presence of
a resonance, it might be confirmed in a decay of the type
${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$, despite the different production
mechanism. The ${{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$ system does not form resonances in the
conventional quark model.

In this paper, a data sample of proton-proton ($pp$) collisions at
$\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$, corresponding to an integrated
luminosity of $5\text{\,fb}^{-1}$, collected by the LHCb experiment in years
of 2016 to 2018, is used to study the decay of
${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$. The ${{\mathchar 28931\relax}^{+}_{c}}$
baryon is reconstructed in the $p{{K}^{-}}{{\pi}^{+}}$ final state. The
${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ decay, with
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$, is chosen as the
normalisation channel, since it has a very similar decay topology and the same
number of tracks in the final state as the signal. This allows a precise
measurement of the ratio of the branching fractions of the
${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
and ${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ decays.

##  2 Detector and signal selection

The LHCb detector [11, 12] 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 (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a 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 the momentum, $p$,
of charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200 GeV. The minimum distance of a track to a primary $pp$
collision vertex (PV), the impact parameter (IP), is measured with a
resolution of $(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in  GeV. The momentum
scale is calibrated using samples of
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}\mu^{-}}$ and
${{{B}^{-}}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ decays
collected concurrently with the data sample used for this analysis [13, 14].
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 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. At the hardware trigger stage, events are required to have a
muon with high $p_{\mathrm{T}}$ or a hadron, photon or electron with high
transverse energy in the calorimeters. The software trigger requires a two-,
three- or four-track secondary vertex with a significant displacement from any
primary $pp$ interaction vertex. Simulation is required to model the effects
of the detector acceptance and the imposed selection requirements. In the
simulation, $pp$ collisions are generated using Pythia 8 [15] with a specific
LHCb configuration [16]. Decays of unstable particles are described by EvtGen
[17], in which final-state radiation is generated using Photos [18]. The
interaction of the generated particles with the detector, and its response,
are implemented using the Geant4 toolkit [19, *Agostinelli:2002hh] as
described in Ref. [21].

The ${{{B}^{-}}}$ candidates are reconstructed with a pair of ${{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}$ baryons along with a companion kaon, where the
${{\mathchar 28931\relax}^{+}_{c}}$ baryon is reconstructed in the
${{\mathchar 28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ final
state. The ${{\mathchar 28931\relax}^{+}_{c}}$ decay products are required to
have good track quality and large transverse momentum, to be inconsistent with
originating from any PV, and to have correct particle identification (PID)
(proton, kaon or pion). The ${{\mathchar 28931\relax}^{+}_{c}}$ candidate is
required to have $p_{\mathrm{T}}$ greater than
$1.8\text{\,Ge\kern-1.00006ptV}$, its vertex inconsistent with originating
from any PV, and the angle between its momentum and its flight direction with
respect to the associated PV to be less than 90 degrees. In case of multiple
primary vertices, the candidate is associated to the PV with respect to which
the smallest $\chi^{2}_{\text{IP}}$ is obtained, where $\chi^{2}_{\text{IP}}$
is defined as the difference in the vertex-fit $\chi^{2}$ of the associated PV
reconstructed with and without the track under consideration. The
reconstructed mass of the ${{\mathchar 28931\relax}^{+}_{c}}$ candidates is
required to be within the window of
$2225<M({p}{{K}^{-}}{{\pi}^{+}})<2345\text{\,Me\kern-1.00006ptV}$. Higher
thresholds on $p_{\mathrm{T}}$ and momentum are set for the companion kaon and
similar requirements as for the ${{\mathchar 28931\relax}^{+}_{c}}$ decay
products are imposed on its track quality, PID and displacement from the PV.
All the selection criteria for the ${{\mathchar 28931\relax}^{+}_{c}}$
candidates apply to the ${{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}$ candidates. The ${{{B}^{-}}}$ candidates are
required to have $p_{\mathrm{T}}$ greater than $5\text{\,Ge\kern-1.00006ptV}$,
a reconstructed lifetime longer than 0.2 ps, and to have a
$\chi^{2}_{\text{IP}}$ less than 25. The reconstructed ${{{B}^{-}}}$ flight
direction, defined by its production and decay vertices, and its momentum are
required to be well aligned. The ${{{B}^{-}}}$ candidate must be within the
mass window of $5205<M({{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}})<5355\text{\,Me\kern-1.00006ptV}$.

To further suppress the background, a boosted decision tree (BDT) [22, 23],
implemented in the TMVA toolkit [24, *TMVA4], is trained with the simulated
sample as signal and with data from sideband regions defined in Table 1 as
background. The training variables are the $\chi^{2}$ per degree of freedom of
the ${{\mathchar 28931\relax}^{+}_{c}}$ decay vertex, and the
$\chi^{2}_{\text{IP}}$ and PID information of all final-state tracks. The PID
values in simulation are drawn from calibration samples to better agree with
real data [26]. The selection requirement on the BDT output is chosen to
optimise the expected significance of the ${{{B}^{-}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$ signal.

Table 1: Definitions of the sideband regions, where $m({{{B}^{-}}})$ and $m({{D}^{+}})$ represent the known values from Ref [27]. The same sideband region is defined for the ${\overline{p}}{{K}^{+}}{{\pi}^{-}}$ combination as for the ${p}{{K}^{-}}{{\pi}^{+}}$ combination. The values are in  MeV. ${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$ |  $50<|M({{\mathchar 28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}})-m({{{B}^{-}}})|<90\text{\,Me\kern-1.00006ptV}$,
---|---
$2225<M({p}{{K}^{-}}{{\pi}^{+}})<2260\text{\,Me\kern-1.00006ptV}$ or
$2310<M({p}{{K}^{-}}{{\pi}^{+}})<2345\text{\,Me\kern-1.00006ptV}$
${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ |  $75<|M({{D}^{-}}{{D}^{+}}{{K}^{-}})-m({{{B}^{-}}})|<120\text{\,Me\kern-1.00006ptV}$,
$45<|M({{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})-m({{D}^{+}})|<90\text{\,Me\kern-1.00006ptV}$

A dedicated BDT is trained for the normalisation channel
${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ with simulation as signal
and with candidates in the ${{D}^{+}}$ and ${{{B}^{-}}}$ sideband regions, as
defined in Table 1, as the proxy for background. The signal region of the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ decay is wider than the
${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
since the energy release ($Q$-value) of the former is larger. The selection is
almost identical to the signal selection, with the ${{\mathchar
28931\relax}^{+}_{c}}$ candidate replaced by the ${{D}^{+}}$ reconstructed
from the ${{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay mode. The ${{D}^{+}}$
candidates are required to be within a mass window of
$1795<M({{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})<1945\text{\,Me\kern-1.00006ptV}$.

##  3 Signal extraction

The ${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
signal is extracted using a three-dimensional (3D) fit to the invariant mass
of ${{{B}^{-}}}$, ${{\mathchar 28931\relax}^{+}_{c}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$
candidates. The 3D fit function is constructed as a sum of multiple terms each
of which is a direct product of one-dimensional functions. The signal function
is $s_{{{{B}^{-}}}}s_{{{\kern 1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}s_{{{\mathchar 28931\relax}^{+}_{c}}}$, where
$s_{{{{B}^{-}}}}$ is modelled by sum of two Gaussian functions sharing common
mean value and $s_{{{\mathchar 28931\relax}^{+}_{c}}}$ is modelled using sum
of three Gaussian functions with a common mean value fixed to the known
${{\mathchar 28931\relax}^{+}_{c}}$ mass [27]. The fit model and parameters
are identical for $s_{{{\mathchar 28931\relax}^{+}_{c}}}$ and $s_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar 28931\relax}}{}^{-}_{c}}}$. The
Gaussian parameters are all fixed to values from fits to simulated samples.
The background shape in the $M({{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}})$ and $M({p}{{K}^{-}}{{\pi}^{+}})$
distributions is parameterised with a first-order polynomial. The background
candidates are of seven different categories: from real ${{{B}^{-}}}$ decay
products, (1) $s^{\prime}_{{{{B}^{-}}}}s_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}b_{{{\mathchar 28931\relax}^{+}_{c}}}$, (2)
$s^{\prime}_{{{{B}^{-}}}}b_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}s_{{{\mathchar 28931\relax}^{+}_{c}}}$, (3)
$s^{\prime}_{{{{B}^{-}}}}b_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}b_{{{\mathchar 28931\relax}^{+}_{c}}}$; from
non-${{{B}^{-}}}$ decays containing true ${\mathchar 28931\relax}_{{c}}$
decays, (4) $b_{{{{B}^{-}}}}s_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}s_{{{\mathchar 28931\relax}^{+}_{c}}}$, (5)
$b_{{{{B}^{-}}}}s_{{{\kern 1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}b_{{{\mathchar 28931\relax}^{+}_{c}}}$, (6)
$b_{{{{B}^{-}}}}b_{{{\kern 1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}s_{{{\mathchar 28931\relax}^{+}_{c}}}$; and pure
combinatorial background, (7) $b_{{{{B}^{-}}}}b_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}b_{{{\mathchar 28931\relax}^{+}_{c}}}$. For
background categories (1)–(3) the $Q$-values of ${{{B}^{-}}}$ decays could
differ from the signal, and a single Gaussian $s^{\prime}_{{{{B}^{-}}}}$ is
used. The fitted ${{{B}^{-}}}$, ${{\mathchar 28931\relax}^{+}_{c}}$ and
${{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$
mass spectra are shown in Fig. 1. The signal yield is determined to be
$1365\pm 42$. It is found that background (4) $b_{{{{B}^{-}}}}s_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}s_{{{\mathchar 28931\relax}^{+}_{c}}}$ makes
negligible contribution; thus the yield is fixed to zero.

LHCb5 fb-1(a)

LHCb5 fb-1(b)

LHCb5 fb-1(c)

Figure 1: Distributions of (a) $M({{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.61993pt\overline{\kern-1.61993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}})$, (b)
$M({\overline{p}}{{K}^{+}}{{\pi}^{-}})$ and (c) $M({p}{{K}^{-}}{{\pi}^{+}})$
of selected ${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.61993pt\overline{\kern-1.61993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
candidates. The data points with error bars are shown along with the total
fitted shape, which is composed of signal and background components, as shown
in the legend.

The resonance structure present in the ${{{B}^{-}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$ decay is studied using the candidates in
narrower ${{{B}^{-}}}$ and ${{\mathchar 28931\relax}^{+}_{c}}$ mass windows:
$5240<M({{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}})<5320\text{\,Me\kern-1.00006ptV}$ and
$2265<M({p}{{K}^{-}}{{\pi}^{+}})<2305\text{\,Me\kern-1.00006ptV}$,
respectively. In order to improve the mass resolution of any resonant state, a
refit of the decay fixing the masses of ${{\mathchar 28931\relax}^{+}_{c}}$
and ${{{B}^{-}}}$ to the known values is performed. The refitted momenta of
${{\mathchar 28931\relax}^{+}_{c}}$, ${{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$ and
${{K}^{-}}$ are used to determine the $M({{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}})$, $M({{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}})$ and $M({{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}})$ mass spectra. It is verified that the selection
requirements do not induce artificial peaking structures. For background
subtraction, it is sufficient to consider the sideband in the two-dimensional
spectrum of $M({p}{{K}^{-}}{{\pi}^{+}})$ and
$M({\overline{p}}{{K}^{+}}{{\pi}^{-}})$, as shown in Fig. 2, since the absence
of the $b_{{{{B}^{-}}}}s_{{{\kern 1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}s_{{{\mathchar 28931\relax}^{+}_{c}}}$ background
component means that whenever a true ${{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$ pair is
selected, it corresponds to a real ${{{B}^{-}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$ decay. The contributions from $s_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}b_{{{\mathchar 28931\relax}^{+}_{c}}}$ and
$b_{{{\kern 1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}s_{{{\mathchar 28931\relax}^{+}_{c}}}$ are averaged
to account for the background contribution in the signal region, where the
double-counted contribution from pure combinatorial background, as in
$b_{{{\kern 1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}b_{{{\mathchar 28931\relax}^{+}_{c}}}$, is
subtracted. The background subtraction is also performed with the $sPlot$
method [28] as a cross-check to verify that none of the observed structures
are due to background fluctuations.

Figure 2: Selected ${{{B}^{-}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.61993pt\overline{\kern-1.61993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$ candidates in the two-dimensional
$M({p}{{K}^{-}}{{\pi}^{+}})$ and $M({\overline{p}}{{K}^{+}}{{\pi}^{-}})$
spectrum. The red dashed box is the signal region, the dark blue, light blue
and green boxes indicate regions dominated by $s_{{{\kern
1.13394pt\overline{\kern-1.13394pt\mathchar
28931\relax}}{}^{-}_{c}}}b_{{{\mathchar 28931\relax}^{+}_{c}}}$, $b_{{{\kern
1.13394pt\overline{\kern-1.13394pt\mathchar
28931\relax}}{}^{-}_{c}}}s_{{{\mathchar 28931\relax}^{+}_{c}}}$ and
combinatorial background, respectively.

The $M({{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}})$ spectrum is shown in Fig.
3. The fitting function is described in detail in Sec. 4. Two states reported
in Ref. [7], ${{\mathchar 28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar
28932\relax}_{c}}(2939)^{0}$, are observed. In addition, a wide structure is
evident around $2880\text{\,Me\kern-1.00006ptV}$. A similar structure was
observed in the prompt ${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$ study [7],
but background due to feed-down from higher ${{\mathchar
28932\relax}^{0}_{c}}$ states, ${{\mathchar 28932\relax}_{c}}(3055)^{0}$ and
${{\mathchar 28932\relax}_{c}}(3080)^{0}$, with a ${{\pi}^{0}}$ or
${{\pi}^{+}}$ missing in the reconstruction, could not be discounted. Such
background will not affect the ${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$
system from ${{{B}^{-}}}$ decays, since there is not enough phase space for
${{{B}^{-}}}\rightarrow{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{\mathchar 28932\relax}_{c}}(3055/3080)^{0}$ and such
feed-down component with a missing particle will not peak at the ${{{B}^{-}}}$
mass. To study the effect of partially reconstructed ${{\mathchar
28932\relax}_{c}}(3055/3080)^{0}$ from other $b$-hadron decays, a sample of
${{\mathchar 28931\relax}^{0}_{b}}$ decays is generated with the decay chain
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28932\relax}_{c}}(3055)^{+}{{D}^{-}}$, ${{\mathchar
28932\relax}_{c}}(3055)^{+}\rightarrow[{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}]_{{\mathchar
28934\relax}_{c}^{++}}{{K}^{-}},{{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$.
The selection criteria remove them completely. Therefore, the wide structure
cannot be due to feed-down, and is considered as a resonant state in the fit.
Both ${{\mathchar 28932\relax}_{c}}(2790)^{0}$ and ${{\mathchar
28932\relax}_{c}}(2815)^{0}$ states are expected at the lower end of the
$M({{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}})$ spectrum, and their masses,
widths and spin-parity $J^{P}$ are fixed to known values [27]. The
${{\mathchar 28932\relax}_{c}}(2790)^{0}$ state has $J^{P}=(1/2)^{-}$ and
${{\mathchar 28932\relax}_{c}}(2815)^{0}$ has $J^{P}=(3/2)^{-}$. However, the
significance of the ${{\mathchar 28932\relax}_{c}}(2815)^{0}$ state is only
$2.1\,\sigma$, and so it is not included in the nominal fit. The yield of the
resonant states is not enough to determine their quantum numbers with
sufficient significance. The spin-parity $J^{P}$ of both ${{\mathchar
28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar 28932\relax}_{c}}(2939)^{0}$
states is fixed to be $(3/2)^{-}$ and that of ${{\mathchar
28932\relax}_{c}}(2880)^{0}$ is fixed to $(1/2)^{-}$ [8].

LHCb5 fb-1

Figure 3: Mass spectrum of the ${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$
pair from the ${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.61993pt\overline{\kern-1.61993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
decays. The data points with error bars are shown along with the total fitted
shape in blue solid line, which is composed of the components, as shown in the
legend.

The invariant mass distributions of the ${{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
and ${{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$ pairs
are shown in Fig. 4. No significant structure is seen. The distributions of
$M({{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}})$ and $M({{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}})$ are also produced after rejecting candidates with
$2900<M({{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}})<2970\text{\,Me\kern-1.00006ptV}$, to remove
contributions from the ${{\mathchar 28932\relax}_{c}}(2923)^{0}$ and
${{\mathchar 28932\relax}_{c}}(2939)^{0}$ states. No significant structure is
seen.

LHCb5 fb-1(a)

LHCb5 fb-1(b)

LHCb5 fb-1(c)

LHCb5 fb-1(d)

Figure 4: Mass spectrum of (a,c) ${{\kern
1.61993pt\overline{\kern-1.61993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
and (b,d) ${{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.61993pt\overline{\kern-1.61993pt\mathchar 28931\relax}}{}^{-}_{c}}$ pairs
from (a,b) all the ${{{B}^{-}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.61993pt\overline{\kern-1.61993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$ candidates and (c,d) after vetoing
candidates with $2900<M({{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}})<2970\text{\,Me\kern-0.90005ptV}$.

##  4 Masses and widths of the excited ${{\mathchar 28932\relax}^{0}_{c}}$
states

The masses and widths of the excited ${{\mathchar 28932\relax}^{0}_{c}}$
states are obtained from an unbinned maximum likelihood fit to the
$M({{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}})$ spectrum. The total fitting
function is constructed as

$f(M)=\frac{N_{{{\mathchar
28932\relax}^{0}_{c}}}}{N_{\mathrm{evt}}}\,f_{{{\mathchar
28932\relax}^{0}_{c}}}(M)\otimes
g(\sigma_{\mathrm{res}})+\frac{N_{\mathrm{phsp}}}{N_{\mathrm{evt}}}\,f_{\mathrm{phsp}}(M)+\frac{N_{\mathrm{bkg}}}{N_{\mathrm{evt}}}\,f_{\mathrm{bkg}}(M),$
(1)

where $M$ stands for $M({{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}})$. The
function describing the excited ${{\mathchar 28932\relax}^{0}_{c}}$ states,
$f_{{{\mathchar 28932\relax}^{0}_{c}}}$, which will be discussed in detail
later, is convolved with a Gaussian function to account for the detector
resolution. The Gaussian function has a mean value of zero and width
$\sigma_{\mathrm{res}}$ in the range of 0–2 MeV dependent on $M({{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}})$. The resolution $\sigma_{\mathrm{res}}$ is
parameterised as
$(M-m_{\mathrm{lo}})^{a}(m_{\mathrm{up}}-M)^{b}[c+d(M-m_{\mathrm{lo}})]$,
where $m_{\mathrm{lo}}=m({{\mathchar 28931\relax}^{+}_{c}})+m({{K}^{-}})$ and
$m_{\mathrm{up}}=m({{{B}^{-}}})-m({{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}})$ are
lower and upper thresholds of the $M({{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}})$ spectrum, respectively, and $a,b,c,d$ are
determined from simulation. The contribution from the non-resonant component
$f_{\mathrm{phsp}}$ is modelled with simulation of pure phase-space decay. The
background contribution $f_{\mathrm{bkg}}$ is extrapolated from the sideband
regions as defined in Fig. 2. The expected number of background events
$N_{\mathrm{bkg}}$ is also fixed using extrapolation from the sideband
regions. The total number of candidates within the fitting range,
$N_{\mathrm{evt}}$, is 1494. All fitting functions are normalised to unity.

As discussed in Sec. 3, four excited states are considered: ${{\mathchar
28932\relax}_{c}}(2790)^{0}$, ${{\mathchar 28932\relax}_{c}}(2880)^{0}$,
${{\mathchar 28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar
28932\relax}_{c}}(2939)^{0}$. The interference between the ${{\mathchar
28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar 28932\relax}_{c}}(2939)^{0}$
states is important and cannot be neglected to correctly describe the data.
The significance of the two-state hypothesis with respect to the hypothesis of
a single ${{\mathchar 28932\relax}_{c}}(2930)^{0}$ state is over $11\,\sigma$.
The function used for excited ${{\mathchar 28932\relax}^{0}_{c}}$ states is
expressed as

$\displaystyle f_{{{\mathchar
28932\relax}^{0}_{c}}}=p\,q\left(\left|\mathcal{M}_{{{\mathchar
28932\relax}_{c}}(2790)^{0}}+\mathcal{M}_{{{\mathchar
28932\relax}_{c}}(2880)^{0}}\right|^{2}+\left|\mathcal{M}_{{{\mathchar
28932\relax}_{c}}(2923)^{0}}+\mathcal{M}_{{{\mathchar
28932\relax}_{c}}(2939)^{0}}\right|^{2}\right)\,,$ (2)

where
$\mathcal{M}_{\alpha}(M)=\hat{A}_{\alpha}(M)F_{l_{\alpha}}(q)F_{L_{\alpha}}(p)$.
The quantities $q$ and $p$ represent the breakup momenta of the ${{\mathchar
28932\relax}^{0}_{c}}$ and ${{{B}^{-}}}$ decays, expressed as
$q=\lambda^{1/2}(M^{2},m_{K}^{2},m_{\Lambda_{c}}^{2})/(2M)$,
$p=\lambda^{1/2}(m_{B}^{2},M^{2},m_{\Lambda_{c}}^{2})/(2M)$, where
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2zx$. The subscript $\alpha$ runs
through all four resonant states considered in the fit. The amplitude of each
state is described by a relativistic Breit–Wigner function,

$\displaystyle\hat{A}_{\alpha}(M)=\frac{c_{\alpha}}{m_{\alpha}^{2}-M^{2}-i\,m_{\alpha}\,\Gamma_{\alpha}\,\frac{q}{q_{\alpha}}\cdot\frac{m_{\alpha}}{M}\cdot\frac{F_{l_{\alpha}}(q)^{2}}{F_{l_{\alpha}}(q_{\alpha})^{2}}},$
(3)

where $m_{\alpha}$ and $\Gamma_{\alpha}$ are the mass and width of the state,
$c_{\alpha}$ is a complex coefficient, and $q_{\alpha}$ is the breakup
momentum $q$ computed at the mass $m_{\alpha}$. The quantities $F_{l}(q)$ and
$F_{L}(p)$ are Blatt–Weisskopf barrier functions, defined as

$\displaystyle F_{k}(q)=\begin{cases}1&k=0,\\\
\sqrt{\frac{(qr)^{2}}{1+(qr)^{2}}}&k=1,\\\
\sqrt{\frac{(qr)^{4}}{9+3(qr)^{2}+(qr)^{4}}}&k=2,\end{cases}$ (4)

where $r$ is the effective radius of the resonant state, fixed to
$3.0\text{\,Ge\kern-1.00006ptV}^{-1}$ [29], $l$ is the orbital angular
momentum between the ${{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}$ and ${{K}^{+}}$ particles, and $L$ is the smallest
possible orbital angular momentum between the ${{\mathchar
28931\relax}^{+}_{c}}$ and ${{\mathchar 28932\relax}^{0}_{c}}$ states. As
mentioned before, the spin-parity of the ${{\mathchar
28932\relax}_{c}}(2790)^{0}$ state is known to be $(1/2)^{-}$ [27]. The 1P
multiplet contains 5 states with the quantum numbers
$(1/2)^{-},(1/2)^{-},(3/2)^{-},(3/2)^{-}$ and $(5/2)^{-}$. The quantum numbers
of the ${{\mathchar 28932\relax}_{c}}(2880)^{0}$ state are assumed to be
$(1/2)^{-}$ as well [8]. The spin-parity of the ${{\mathchar
28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar 28932\relax}_{c}}(2939)^{0}$
states is assumed to be $(3/2)^{-}$. The values of $l$ are set to respect
momentum and parity conservation, namely $l=0$ for the ${{\mathchar
28932\relax}_{c}}(2790)^{0}$ and ${{\mathchar 28932\relax}_{c}}(2880)^{0}$
states and $l=2$ for the ${{\mathchar 28932\relax}_{c}}(2923)^{0}$ and
${{\mathchar 28932\relax}_{c}}(2939)^{0}$ states. Alternative assumptions for
the quantum numbers of the states are considered in the determination of the
systematic uncertainty. If the spin of the ${{\mathchar 28932\relax}^{0}_{c}}$
of interest is $J$, then $L$ can only take values $J-1/2$ and $J+1/2$, and
$F_{L_{\alpha}}$ is replaced by
$F_{J_{\alpha}-1/2}+k_{\alpha}F_{J_{\alpha}+1/2}$. The $k_{\alpha}$ parameters
are complex factors representing the contribution of the higher partial waves.
They cannot be extracted from a one-dimensional fit with the available
statistics, and all $k_{\alpha}$ parameters are set to a common value of
$k_{0}$ in the fit.

To study the possible bias on the measured mass and width of the ${{\mathchar
28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar 28932\relax}_{c}}(2939)^{0}$
states, $3\,000$ pseudoexperiments are performed where all other parameters,
except the masses and widths, are fixed. A fit is performed for each
pseudoexperiment, and the pull of each mass or width parameter is calculated
with respect to the input. The pull is defined as the difference between the
fitted value and the input value, divided by the uncertainty obtained from the
fit. The pull distributions are then fitted with Gaussian functions. The
deviation of the Gaussian mean from zero is used to correct the fitted mass
values. The correction values are smaller than the statistical uncertainties
and will be considered in the systematic uncertainty determination.

The fitted $M({{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}})$ distribution is
shown in Fig. 3, and the measured masses and widths are listed in Table 2. The
significance of the ${{\mathchar 28932\relax}_{c}}(2790)^{0}$ and ${{\mathchar
28932\relax}_{c}}(2880)^{0}$ states is calculated by studying $30\,000$
pseudoexperiments. Each is generated with a null hypothesis, then fitted both
with and without the excited ${{\mathchar 28932\relax}^{0}_{c}}$ state of
interest. The test statistic $t_{0}$, defined as twice the difference in log-
likelihood with and without the state,
$2\log(\mathcal{L}_{1}/\mathcal{L}_{0})$, is expected to follow a $\chi^{2}$
distribution. The $t_{0}$ values from the pseudoexperiments are fitted with a
$\chi^{2}$ distribution, and the $p$-value of the observed yield corresponds
to the fraction of integrated area above the $t_{0}$ value measured in real
data divided by the total integrated area. The significance of both the
${{\mathchar 28932\relax}_{c}}(2790)^{0}$ and ${{\mathchar
28932\relax}_{c}}(2880)^{0}$ states is estimated to be $3.9\,\sigma$.

Table 2: Measured masses, widths and significance of excited ${{\mathchar 28932\relax}^{0}_{c}}$ states.  State | Mass (MeV) | Width (MeV) | Significance
---|---|---|---
${{\mathchar 28932\relax}_{c}}(2880)^{0}$ | $2881.8\pm 3.1\pm 8.5$ | $12.4\pm 5.2\pm 5.8$ | $>10\sigma$
${{\mathchar 28932\relax}_{c}}(2923)^{0}$ | $2924.5\pm 0.4\pm 1.1$ | $\phantom{0}4.8\pm 0.9\pm 1.5$ | $>10\sigma$
${{\mathchar 28932\relax}_{c}}(2939)^{0}$ | $2938.5\pm 0.9\pm 2.3$ | $11.0\pm 1.9\pm 7.5$ | $3.8\sigma$

Systematic uncertainties on the mass and width measurements from various
sources are studied. Multiple alternative assumptions on the fixed parameters
are tested. The spin-parity of the ${{\mathchar 28932\relax}_{c}}(2923)^{0}$
and ${{\mathchar 28932\relax}_{c}}(2939)^{0}$ states is set to $(1/2)^{-}$,
$(1/2)^{+}$ or $(3/2)^{-}$, and $J^{P}$ of the ${{\mathchar
28932\relax}_{c}}(2880)^{0}$ is set to $(1/2)^{+}$, $(3/2)^{-}$ or
$(3/2)^{+}$. In these tests, the states with the same spin-parity are always
added coherently. The effective radius $r$ is set to either 2.0 or
$4.0\text{\,Ge\kern-1.00006ptV}^{-1}$. The mass and width of the ${{\mathchar
28932\relax}_{c}}(2790)^{0}$ are varied within their uncertainty. A different
coefficient $k_{\alpha}$ is assigned to each group of ${{\mathchar
28932\relax}^{0}_{c}}$ states with the same spin-parity. An additional state
around 2970 MeV with orbital angular momentum of 0, 1 or 2 is added. The fit
including the ${{\mathchar 28932\relax}_{c}}(2815)^{0}$ state is considered.
The potential interference with non-resonant decays is considered by adding a
constant term in the ${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$ mass
distribution. The maximum variation in the fit results is obtained for each
alternative assumption if multiple values are considered, and the total
systematic uncertainty due to model assumptions is the quadratic sum of
variation in all alternative assumptions.

The ${{\mathchar 28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar
28932\relax}_{c}}(2939)^{0}$ lineshape is described alternatively using a
K-matrix formalism [30], which preserves unitarity. The variation in the
results is considered as a systematic uncertainty. The pull distributions from
pseudoexperiments are used to correct a possible bias in the masses and widths
of the ${{\mathchar 28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar
28932\relax}_{c}}(2939)^{0}$ states, where the resulting corrections are
smaller than statistical uncertainties. It is found that the fitted width
$\sigma_{\rm{pull}}$ of the pull distribution is slightly wider than unity,
indicating a potential under-estimation of the statistical uncertainty.
Therefore, the term
$\sqrt{{\sigma}^{2}_{\rm{pull}}-1}\cdot\sigma_{\mathrm{stat}}$ is assigned as
a systematic uncertainty. A relative uncertainty of $3\times 10^{-4}$ on the
charged particle momentum scale [7] is propagated through the simulation. The
resolution $\sigma_{\mathrm{res}}$ is varied within uncertainty and the
resulting difference on the fitted parameters of the resolution function is
then propagated to the masses and natural widths of the excited ${{\mathchar
28932\relax}^{0}_{c}}$ states, which is found to be negligible. The magnitude
of the energy-loss correction for charged particles is known to $10$% accuracy
[31]. In the study of Ref. [14], a correction of less than
$0.01\text{\,Me\kern-1.00006ptV}$ per track is estimated. A conservative
estimation of $0.01\text{\,Me\kern-1.00006ptV}$ per track is taken as the
systematic uncertainty, which is also found to be negligible.

Mass constraints on the ${{{B}^{-}}}$ and ${{\mathchar 28931\relax}^{+}_{c}}$
candidates are applied when calculating $M({{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}})$. The maximum change after varying the
${{{B}^{-}}}$ or ${{\mathchar 28931\relax}^{+}_{c}}$ mass by one standard
deviation is assigned as a systematic uncertainty. The systematic uncertainty
due to the background is estimated by generating and fitting 300
pseudoexperiments, randomly varying the background yield according to a
Poisson distribution while fixing the ${{\mathchar 28932\relax}^{0}_{c}}$
yield, the parameters of the ${{\mathchar 28932\relax}^{0}_{c}}$ model and the
non-resonant yield. The average of the fitted values is calculated, and its
difference from the nominal result is assigned as a systematic uncertainty.

A summary of the systematic uncertainties are listed in Table 3. The
uncertainties from different sources are uncorrelated and are added in
quadrature to give the total systematic uncertainty. The significance of the
${{\mathchar 28932\relax}_{c}}(2880)^{0}$ and ${{\mathchar
28932\relax}_{c}}(2790)^{0}$ states is $3.8\,\sigma$ and $3.7\,\sigma$,
respectively, after taking the systematic uncertainties into account.

Table 3: Systematic uncertainties on the masses and widths of the ${{\mathchar 28932\relax}_{c}}(2923)^{0}$, ${{\mathchar 28932\relax}_{c}}(2939)^{0}$ and ${{\mathchar 28932\relax}_{c}}(2880)^{0}$ states. Values are given in  MeV. | ${{\mathchar 28932\relax}_{c}}(2923)^{0}$ | ${{\mathchar 28932\relax}_{c}}(2939)^{0}$ | ${{\mathchar 28932\relax}_{c}}(2880)^{0}$
---|---|---|---
Source | Mass | Width | Mass | Width | Mass | Width
Model assumption | 0.8 | 1.4 | 1.9 | 7.0 | 8.4 | 4.1
Lineshape formalism | 0.7 | 0.1 | 1.3 | 0.8 | 1.1 | 0.0
Bias correction | 0.2 | 0.4 | 0.4 | 2.1 | $-$ | $-$
Momentum scale | 0.0 | $-$ | 0.1 | $-$ | 0.0 | $-$
Mass constraint | 0.1 | 0.1 | 0.1 | 0.5 | 0.1 | 0.4
Background | 0.1 | 0.3 | 0.2 | 0.8 | 0.7 | 4.0
Total | 1.1 | 1.5 | 2.3 | 7.5 | 8.5 | 5.8

##  5 Branching fraction

The relative branching fraction $R_{{\mathcal{B}}}$ of the
${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
decay with respect to the ${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$
channel is calculated as a ratio of efficiency-corrected yield divided by the
corresponding charm decay branching fractions, expressed as

$\displaystyle R_{{\mathcal{B}}}=\frac{N({{{B}^{-}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}})}{N({{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}})}\times\frac{{\varepsilon_{\mathrm{tot}}}({{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}})}{{\varepsilon_{\mathrm{tot}}}({{{B}^{-}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}})}\times\frac{{\mathcal{B}}^{2}({{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})}{{\mathcal{B}}^{2}({{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}})}.$ (5)

The yield of the signal and normalisation channels is obtained from the 3D fit
described in Sec. 3. The charm decay branching fractions
${\mathcal{B}}({{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})$
[$(9.38\pm 0.16)$%] and ${\mathcal{B}}({{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}})$ [$(6.28\pm 0.32)$%]
are obtained from Ref. [27].

In addition to the selection described in Sec. 2, it is required that all
selected signal events are triggered on the ${{{B}^{-}}}$ candidate decay
products for both signal and normalisation modes so that the trigger
efficiencies can be estimated accurately. The projections of the 3D fit to the
${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
spectra with these additional requirements are shown in Fig. 5 (a-c) and the
measured yield is $977\pm 36$. For the normalisation decay
${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$, a similar 3D fit is
performed to the mass distributions of the ${{{B}^{-}}}$, ${{D}^{+}}$ and
${{D}^{-}}$ candidates with the same fitting functions except replacing
${\mathchar 28931\relax}_{{c}}$ with $D$ wherever applicable. The fitted
distributions are shown in Fig. 5 (d-f), and the yield of
${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ decay is $2212\pm 62$.

LHCb5 fb-1(a)

LHCb5 fb-1(b)

LHCb5 fb-1(c)

LHCb5 fb-1(d)

LHCb5 fb-1(e)

LHCb5 fb-1(f)

Figure 5: Distributions of (a) $M({{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.61993pt\overline{\kern-1.61993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}})$, (b)
$M({\overline{p}}{{K}^{+}}{{\pi}^{-}})$ and (c) $M({p}{{K}^{-}}{{\pi}^{+}})$
in the ${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.61993pt\overline{\kern-1.61993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
sample and distributions of (d) $M({{D}^{-}}{{D}^{+}}{{K}^{-}})$, (e)
$M({{K}^{+}}{{\pi}^{-}}{{\pi}^{-}})$ and (f)
$M({{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})$ in the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ sample. All selected
events are triggered due to ${{{B}^{-}}}$ candidates. The legend is the same
as in Fig. 1, except replacing ${{\mathchar 28931\relax}^{+}_{c}}$ (${{\kern
1.61993pt\overline{\kern-1.61993pt\mathchar 28931\relax}}{}^{-}_{c}}$) with
${{D}^{+}}$ (${{D}^{-}}$) for the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ decay.

The total efficiency ${\varepsilon_{\mathrm{tot}}}$ is the product of the
efficiencies due to the detector acceptance, trigger, reconstruction, offline
preselection and multivariate selection. The efficiencies from different
sources are estimated with a combination of simulation and data. Weights are
assigned to the simulated events such that the track multiplicity distribution
agrees with the data. The tracking efficiency for charged tracks is estimated
using data, and the difference between simulation and data is obtained from a
correction table as a function of $p_{\mathrm{T}}$, $\eta$ and number of
charged tracks [32]. The total efficiencies of the signal and normalisation
decays are $(3.41\pm 0.02)\times 10^{-4}$ and $(8.15\pm 0.11)\times 10^{-4}$,
respectively, where the uncertainties come from the statistics of the
simulation samples. An alternative set of selections of the
${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
signal with tighter criteria on $p_{\mathrm{T}}$ and looser criteria on
${{{B}^{-}}}$ decay time is used as a cross-check, and the efficiency-
corrected yield is found to be consistent with the baseline selections.

The systematic uncertainty on the relative branching fraction comes from
uncertainties in signal yield determination and efficiency estimation. The
working point of the BDT output is chosen to maximise the signal significance.
The choice of BDT thresholds are varied and no significant bias is found. The
average of the efficiency-corrected yield agrees with the nominal result
within statistical uncertainties, and the maximum change in the central value
is taken as a systematic uncertainty. Alternative assumptions are made when
performing the 3D fits, including varying the fitting range, using fixed
shapes from simulation for the signal in the fit, and varying the mass
resolution for $B$ decay backgrounds. The shifts in yield are taken as
systematic uncertainties. The simulation sample is weighted to match the track
multiplicity distribution of data, and the uncertainties of the weights are
propagated to the efficiencies. The efficiency dependence on the decay phase
space is studied in the $M({{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}})$ vs. $M({{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}})$ ($M({{D}^{-}}{{K}^{-}})$ vs.
$M({{D}^{+}}{{K}^{-}})$) plane for the signal (normalisation) decay. The
efficiencies are calculated alternatively according to each candidate’s decay
phase space and the shift in central values is taken as a systematic
uncertainty. The tracking efficiency has a negligible effect on the final
results. The PID values in simulation are sampled to better agree with data
using a dedicated tool [26]. An alternative template is used for the sampling
and the shift in the efficiency-corrected yield is taken as a systematic
uncertainty. The parameterisation of the VELO materials, which influences IP-
related variables, is tuned on simulation. The $\chi^{2}_{\text{IP}}$
distribution in simulation is scaled with parameters given by the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{-}}$
sample, and the shift in $R_{{\mathcal{B}}}$ using the scaled simulation is
assigned as a systematic uncertainty. The trigger efficiencies are estimated
using the simulation as the baseline. They are also estimated using data,
taking the events triggered independent of the ${{{B}^{-}}}$ candidates. The
estimated efficiencies between the two methods agree within statistical
uncertainties, and the difference between efficiencies calculated using data
and the nominal value is assigned as a systematic uncertainty. The uncertainty
due to the limited simulation sample size is propagated to the final result.
The systematic uncertainty on $R_{{\mathcal{B}}}$ is summarised in Table 4,
and all the terms mentioned above are added in quadrature.

The ratio of branching fractions is measured to be

$\displaystyle R_{{\mathcal{B}}}=2.36\pm 0.11\pm 0.22\pm 0.25,$

where the uncertainties are statistical, systematic and that due to
uncertainties on the ${{\mathchar 28931\relax}^{+}_{c}}$ and ${{D}^{+}}$
branching fractions. This measurement is significantly improved with respect
to $R_{{\mathcal{B}}}=2.23\pm 0.78$ calculated using values from Ref. [27].

Table 4: Relative systematic uncertainties of the branching fractions of signal, normalisation channel, and their ratio $R_{{\mathcal{B}}}$ (in percent). Correlation between the two channels are considered. Source | ${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$ | ${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ | $R_{{\mathcal{B}}}$
---|---|---|---
BDT working point | 1.5 | 0.4 | 1.6
Fit range | 1.2 | 0.6 | 1.3
Detector resolution | 0.0 | 0.3 | 0.3
Signal model | 0.0 | 1.6 | 1.6
Simulation weighting | 2.6 | 3.8 | 4.6
Decay phase space | 0.1 | 0.2 | 0.3
PID resample | 0.6 | 0.9 | 1.0
Detector parameterisation | 3.3 | 2.5 | 0.8
Trigger | 5.6 | 4.9 | 7.5
Simulation sample size | 0.9 | 1.2 | 1.5
Total |  |  | 9.4

##  6 Summary

The ${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
decay is studied for the first time in $pp$ collisions using data collected by
the LHCb experiment, corresponding to an integrated luminosity of
$5\text{\,fb}^{-1}$. In the ${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$
invariant mass spectrum, two neutral excited charm baryon states, ${{\mathchar
28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar 28932\relax}_{c}}(2939)^{0}$,
are observed. Their masses and widths are in agreement with those of the
states observed in a prompt ${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$
measurement [7]. These new measurements confirm that the ${{\mathchar
28932\relax}_{c}}(2930)^{0}$ state observed in
${{{B}^{-}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{-}}$
at $B$-factories is resolved into two separate states. No resonance structure
is found at higher masses. Evidence for the ${{\mathchar
28932\relax}_{c}}(2880)^{0}$ state is found with a significance of
$3.8\,\sigma$. In addition, evidence of a new ${{\mathchar
28932\relax}_{c}}(2790)^{0}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}$ decay is found with a significance of
$3.7\,\sigma$.

The relative branching fraction of ${{{B}^{-}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{-}}$ is measured with respect to
${{{B}^{-}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{-}}$ as
$R_{{\mathcal{B}}}=2.36\pm 0.11\pm 0.22\pm 0.25$, where the uncertainties are
statistical, systematic and that due to the charm decay branching fraction.
This is the most precise measurement to date of this ratio.

## References

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  * [3] Belle collaboration, Y. B. Li et al., _Evidence of a structure in $\bar{K}^{0}{{\mathchar 28931\relax}^{+}_{c}}$ consistent with a charged ${{\mathchar 28932\relax}_{c}}(2930)^{+}$ , and updated measurement of $\bar{B}^{0}\rightarrow\bar{K}^{0}{{\mathchar 28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$ at Belle_, Eur. Phys. J. C78 (2018) 928, arXiv:1806.09182
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The Einstein and Møller energy-momentum complexes in post-Newtonian
approximation
I-Ching Yang<EMAIL_ADDRESS>
Department of Applied Science, National Taitung University,

Taitung 95002, Taiwan (R.O.C.)

ABSTRACT

In the first and second post-Newtonian approximation of the Schwarzschild
metric, I obtain the energy component of the Einstein and Møller energy-
momentum complex. Both energies involve the rest-mass energy $m$, the energy
stored in the configuration and that in the gravitational field, but the
energies of Schwarzschild spacetime in the Einstein and Møller prescriptions
are the total mass-energy $M$. First, for general relativity, the rest-mass
energy $m$ in the flat spacetime behaves like the bare mass, and the total
mass-energy $M$ in the curved spacetime behaves like the experimentally
observed mass. Second, the zero-potential surface is important condition for
defining the energy of gravitational field, and plays an important role in the
energy-momentum localization of general relativity.

In the evolution of physics, its unity is maintained by a network of
correspondence principles, through which simpler theories maintain their
vitality by links to more sophisticated but more accurate ones. The
correspondence between the newer theory and its predecessor gives one the
ability to recover the older theory from the newer. In view of Newton’s theory
of gravitation which is inconsistent with special relativity, Einstein was
published general relativity based on equivalence principle, where the
gravitational field is described by a metric. Furthermore, general relativity
will reduce to Newton’s theory of gravity in the limit of weak gravitational
fields and low velocities. Einstein [1] built the Newtonian limit into general
relativity, and also computed the precession of the perihelion which is one of
the post-Newtonian effects. The formalism of Newtonian theory plus post-
Newtonian corrections is called the “post-Newtonian approximation” [2].

One of the most important themes of general relativity is the energy-momentum
localization which has no satisfactory solution. At present there is not an
accepted definition of the localized energy-momentum associated with the
gravitational field. Nevertheless, several approaches have been carried out to
study the energy-momentum localization. A number of definitions for energy-
momentum complexes in general relativity have been given by many authors,
including Einstein [3], Møller [4], Tolman [5], Papapetrou [6], Bergmann and
Thomson [7], Landau and Lifshitz [8], and Weinberg [9]. Energy-momentum
complexes are coordinate-dependent pseudotensorial quantities, except Møller,
which can be used in quasi-Cartesian coordinates, more precisely in
Schwarzschild Cartesian coordinates [10], in Kerr-Schild Cartesian coordinates
[11], and in generalized Painlevé-Gullstrand Cartesian coordinates [12]. Yang
et al. [13] have shown that the energy component of the energy-momentum
complexes in Einstein prescription $E_{\rm E}$ and in Møller prescription
$E_{\rm M}$ will be unlike beside vacuum case $T^{0}_{0}=0$ earlier. Then,
Vagenas [14], Matyjasek [15] and Yang [12] attempt to find out the relation
formula about $E_{\rm E}$ and $E_{\rm M}$. In this article, according to the
post-Newtonian approximation, I will study the difference between $E_{\rm E}$
and $E_{\rm M}$, and their relationship.

In order to obtain conserved quantities, Einstein introduce the energy-
momentum complex

$\Theta^{\mu}_{\nu}=\sqrt{-g}\left(T^{\mu}_{\nu}+t^{\mu}_{\nu}\right),$ (1)

which satisfies the differential conservation form
$\partial_{\mu}\Theta^{\mu}_{\nu}=0$. Here $T^{\mu}_{\nu}$ is the energy-
momentum tensor of matter and $t^{\mu}_{\nu}$ is the energy-momentum
pseudotensor from the gravitational field. Hence, an antisymmetric ${\cal
U}^{\mu\rho}_{\nu}$ in the indices $\mu$ and $\rho$ could be introduced
mathematically as

$\Theta^{\mu}_{\nu}\equiv\frac{\partial{\cal U}^{\mu\rho}_{\nu}}{\partial
x^{\rho}},$ (2)

which is called “superpotential”. In the Einstein prescription, the energy-
momentum complex is

$\Theta^{\mu}_{\nu}=\frac{1}{16\pi}\frac{\partial H^{\mu\rho}_{\nu}}{\partial
x^{\rho}},$ (3)

with Freud’s superpotential

$H^{\mu\rho}_{\nu}=\frac{g_{\nu\sigma}}{\sqrt{-g}}\left[(-g)\left(g^{\mu\sigma}g^{\rho\alpha}-g^{\mu\alpha}g^{\rho\sigma}\right)\right],$
(4)

and in the Møller prescription, the energy-momentum complex reads

$\Theta^{\mu}_{\nu}=\frac{1}{8\pi}\frac{\partial\chi^{\mu\rho}_{\nu}}{\partial
x^{\rho}},$ (5)

with Møller’s superpotential

$\chi^{\mu\rho}_{\nu}=\sqrt{-g}\left(\frac{\partial g_{\nu\alpha}}{\partial
x^{\beta}}-\frac{\partial g_{\nu\beta}}{\partial
x^{\alpha}}\right)g^{\mu\beta}g^{\rho\alpha}.$ (6)

Thus, the energy within the chosen region $\Sigma$ would be shown to be

$E=\int_{\Sigma}\Theta^{0}_{0}d^{3}x.$ (7)

In the weak-field case, for the solar system, the Schwarzschild metric can be
expanded to give the first post-Newtonian form

$ds^{2}=(1-2\phi)dt^{2}-(1+2\phi)(dx^{2}+dy^{2}+dz^{2}).$ (8)

Here $\phi$ is the Newton’s gravitational potential, and normalized such that
$\phi(\infty)=0$. To begin with, the corresponding Freud superpotential
$H^{0i}_{0}$ is obtained

$H^{0i}_{0}=4\sqrt{\frac{1-2\phi}{1+2\phi}}\partial_{i}\left(\frac{1+2\phi}{1-2\phi}\right)\frac{x^{i}}{r},$
(9)

and be expanded in powers of $\phi$ as

$H^{0i}_{0}=\left[4\partial_{i}\phi+{\cal{O}}(\phi^{2})\right]\frac{x^{i}}{r}.$
(10)

The energy component of the Einstein energy-momentum complex is evaluated to
be

${}_{E}\Theta^{0}_{0}=\frac{1}{4\pi}\nabla^{2}\phi,$ (11)

and these higher-order terms ${\cal{O}}(\phi^{2})$ is ignored. Eventually, in
the Einstein prescription, the energy within the region $\Sigma$ that includes
mass distribution is given by

$E_{\rm E}=\frac{1}{4\pi}\int_{\Sigma}\nabla^{2}\phi d^{3}x.$ (12)

Since the gravitational potential satisfes the field equation with a matter
density $\rho$

$\nabla^{2}\phi=4\pi\rho$ (13)

and the total mass is defined by

$\int_{\Sigma}\rho d^{3}x=m.$ (14)

Hence, the energy in the Einstein prescription can be rewritten to

$E_{\rm E}=m.$ (15)

Subsequently, the corresponding Møller superpotential is found to be

$\chi^{0i}_{0}=-\sqrt{\frac{1+2\phi}{1-2\phi}}\partial_{i}(1-2\phi)\frac{x^{i}}{r},$
(16)

which is expanded in power of $\phi$ as

$\chi^{01}_{0}=\left[2\partial_{i}\phi+{\cal{O}}(\phi^{2})\right]\frac{x^{i}}{r}.$
(17)

The energy component of the Møller energy-momentum complex is described to

${}_{M}\Theta^{0}_{0}=\frac{1}{4\pi}\nabla^{2}\phi+{\cal{O}}(\phi^{2}).$ (18)

After the higher-order terms ${\cal{O}}(\phi^{2})$ is ignored, the energy in
the Møller prescription with the region $\Sigma$ is exhibited

$E_{\rm M}=m.$ (19)

The term of the right-hand side in Eq.(15) and Eq.(19) is the “rest-mass
energy” as the invariant mass is the rest energy in special relativity.

Next, the second post-Newtonian form of the Schwarzschild metric is considered
for post-Newtonian corrections to the Newtonian treatment [9]

$\begin{split}ds^{2}=&(1-2\phi+2\phi^{2})dt^{2}-(1+2\phi+\phi^{2})(dx^{2}+dy2+dz^{2})\\\
&-\frac{\phi^{2}}{r^{2}}(xdx+ydy+zdz)^{2}.\end{split}$ (20)

Therefore, the Freud superpotential is given

$H^{0i}_{0}=\left[4\partial_{i}\phi+2\phi\partial_{i}\phi+\frac{2\phi^{2}}{r}+{\cal{O}}(\phi^{3})\right]\frac{x^{i}}{r}.$
(21)

The energy component of the Einstein energy-momentum complex is

${}_{E}\Theta^{0}_{0}=\frac{1}{4\pi}\nabla^{2}\phi+\frac{1}{8\pi}[\phi\nabla^{2}\phi+(\nabla\phi)^{2}]+\frac{1}{8\pi}\nabla\cdot\frac{\phi^{2}}{r}\hat{r},$
(22)

and the energy in the Einstien prescription can be shown

$E_{\rm E}=m+\frac{1}{2}\int_{\Sigma}\rho\phi
d^{3}x+\frac{1}{8\pi}\int_{\Sigma}(\nabla\phi)^{2}d^{3}x+\frac{1}{8\pi}\oint_{\partial\Sigma}\frac{\phi^{2}}{r}\hat{r}\cdot
d\vec{a}.$ (23)

Here, the second term of Eq.(23) represents the energy stored in the
configuration

$E_{\rm config}\equiv\frac{1}{2}\int_{\Sigma}\rho\phi d^{3}$ (24)

and the third term represents the energy stored in the gravitational field

$E_{\rm field}\equiv\frac{1}{8\pi}\int_{\Sigma}(\nabla\phi)^{2}d^{3}x.$ (25)

Thus, the energy in the Einstien prescription can be rewritten to

$E_{\rm E}=m+E_{\rm config}+E_{\rm
field}+\frac{1}{8\pi}\oint_{\partial\Sigma}\frac{\phi^{2}}{r}\hat{r}\cdot
d\vec{a}.$ (26)

Afterward the corresponding Møller superpotential is found

$\chi^{01}_{0}=[2\partial_{i}\phi-4\phi\partial_{i}\phi+{\cal{O}}(\phi^{3})]\frac{x^{i}}{r}.$
(27)

The energy component of the Møller energy-momentum complex is described as

${}_{M}\Theta^{0}_{0}=\frac{1}{4\pi}\nabla^{2}\phi-\frac{1}{2\pi}\left[\phi\nabla^{2}\phi+(\nabla\phi)^{2}\right]+{\cal{O}}(\phi^{3})$
(28)

To ignore the higher-order terms ${\cal{O}}(\phi^{3})$, the energy in the
Møller prescription with the region $\Sigma$ is exhibited as

$E_{\rm M}=m-4E_{\rm config}-4E_{\rm field}$ (29)

In summary, I obtain the energy of the Schwarzschild spacetime in the first
post-Newtonain approximation by using the Einstein and Møller energy-momentum
complexes, and the results can be rewritten as

$E_{\rm E}=m+\left(E_{\rm config}+E_{\rm
field}\right)+\frac{1}{8\pi}\oint_{\partial\Sigma}\frac{\phi^{2}}{r}\hat{r}\cdot
d\vec{a}$ (30)

and

$E_{\rm M}=m-4\left(E_{\rm config}+E_{\rm field}\right).$ (31)

First, both energies $E_{\rm E}$ and $E_{\rm M}$ involve the rest-mass energy
$m$, the energy stored in the configuration $E_{\rm config}$ and that in the
gravitational field $E_{\rm field}$ and all terms can be realized in special
relativity and Newton’s theory of gravity. However, in Schwarzschild
spacetime, the energies in the Einstein and Møller prescriptions are

$\bar{E}_{\rm E}=\bar{E}_{\rm M}=M,$ (32)

which $M$ is the “total mass-energy” [2] . Referring to quantum
electrodynamics, due to vacuum polarization, bare mass and bare charge of
electrons are not measurable. Hence, for general relativity, the rest-mass
energy $m$ in the flat spacetime behaves like bare mass, and the total mass-
energy $M$ in the curved spacetime behaves like the experimentally observable
mass. The total mass-energy $M$ can be regarded as a modified $m$. Second, the
difference of energy between the Einstein and Møller prescriptions [13] is
defined as

$\Delta E=E_{\rm E}-E_{\rm M},$ (33)

giving

$\Delta E=\frac{5}{8\pi}\oint_{\partial\Sigma}\phi\nabla\phi\cdot
d\vec{a}+\frac{1}{8\pi}\oint_{\partial\Sigma}\frac{\phi^{2}}{r}\hat{r}\cdot
d\vec{a}.$ (34)

While gravitational potential $\phi=0$ or gravitational field $\nabla\phi=0$
in the region $\partial\Sigma$, the difference of energy $\Delta E$ will
become zero. Let us turn our attention to the calculation of potential energy,
the definition of zero-potential surface is primarily important condition. In
the Einstein prescription, the first derivative of the metric
$g_{\mu\nu,\alpha}$ was required to be zero, so the zero-potential surface of
the Einstein energy-momentum complex is a flat spacetime. However, there is no
manifest condition in the Møller prescription, and the zero-potential surface
of the Møller energy-momentum complex can not be described. The zero-potential
surface will be an important role in the energy-momentum localization of
general relativity and should be considered seriously. Meanwhile, the choice
of reference in the boundary term of the quasi-local energy expressions [16]
and the role of thermodynamic potential [12, 17], could be associated with the
quasi-localized energy.

Acknowledgments

I would like to thank Prof. Yaw-Hwang Chen for discussions and Mr. Yi-Hsun Li
for partial calculation.

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