EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-226 LHCb-PAPER-2014-048 Sept. 30, 2014

Precision measurement of the mass and lifetime of the ${{\mathchar
28932\relax}^{-}_{b}}$ baryon

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

We report on measurements of the mass and lifetime of the ${{\mathchar
28932\relax}^{-}_{b}}$ baryon using about 1800 ${{\mathchar
28932\relax}^{-}_{b}}$ decays reconstructed in a proton-proton collision data
set corresponding to an integrated luminosity of 3.0$\mbox{\,fb}^{-1}$
collected by the LHCb experiment. The decays are reconstructed in the
${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{c}}{{\pi}^{-}}$, ${{\mathchar
28932\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ channel and
the mass and lifetime are measured using the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{-}}$
mode as a reference. We measure

$\displaystyle M({{\mathchar 28932\relax}^{-}_{b}})-M({{\mathchar
28931\relax}^{0}_{b}})$ $\displaystyle=178.36\pm 0.46\pm
0.16~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$
$\displaystyle\frac{\tau_{{{\mathchar
28932\relax}^{-}_{b}}}}{\tau_{{{\mathchar 28931\relax}^{0}_{b}}}}$
$\displaystyle=1.089\pm 0.026\pm 0.011,$

where the uncertainties are statistical and systematic, respectively. These
results lead to a factor of two better precision on the ${{\mathchar
28932\relax}^{-}_{b}}$ mass and lifetime compared to previous best
measurements, and are consistent with theoretical expectations.

Published in Physical Review Letters

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

Over the last two decades, beauty mesons have been studied in detail. Various
theoretical approaches allow one to relate measured decay rates to Standard
Model parameters. One of the most predictive tools is the heavy quark
expansion (HQE) [1, 2, 3, 4, 5, 6, 7, 8], which describes the decay rates of
beauty hadrons through an expansion in powers of $\Lambda_{\rm QCD}/m_{b}$,
where $\Lambda_{\rm QCD}$ is the energy scale at which the strong-interaction
coupling becomes large, and $m_{b}$ is the $b$-quark mass. In addition to the
total $b$-hadron decay widths, HQE can be used to calculate $b$-hadron
parameters required for the measurement of coupling strengths between quarks
in charged-current interactions, which in turn provides constraints on physics
beyond the Standard Model.

A stringent test of HQE is to confront its predictions for lifetimes, i.e.,
the inverse of the corresponding decay widths, with precision measurements.
The lifetimes of the ${{B}^{0}}$ and ${{{B}^{+}}}$ mesons are measured to a
precision of about 0.5% [9], the ${{B}^{0}_{s}}$ meson to 1% [10, 9], and the
${{\mathchar 28931\relax}^{0}_{b}}$ baryon to 0.7% [9], and their values are
in agreement with HQE predictions [11].

Another interesting test is to compare the measured lifetime ratio
$\tau({{\mathchar 28932\relax}^{-}_{b}})/\tau({{\mathchar
28932\relax}^{0}_{b}})$ to HQE predictions. Since penguin contraction terms
cancel in this ratio [12], a more precise prediction is possible compared to
$\tau({{\mathchar 28931\relax}^{0}_{b}})/\tau({{B}^{0}})$. One prediction
leads to $\tau({{\mathchar 28932\relax}^{-}_{b}})/\tau({{\mathchar
28932\relax}^{0}_{b}})=1.05\pm 0.07$ [12], where the dominant uncertainties
are related to matrix elements that are calculable using lattice quantum
chromodynamics (QCD) [13]. A phenomenological analysis of the relevant matrix
elements using charm baryon lifetimes leads to a prediction of
$1/\tau({{\mathchar 28931\relax}^{0}_{b}})-1/\tau({{\mathchar
28932\relax}^{-}_{b}})=0.11\pm 0.03$ ps-1 [14], or $\tau({{\mathchar
28932\relax}^{-}_{b}})/\tau({{\mathchar
28931\relax}^{0}_{b}})=1.19^{+0.07}_{-0.06}$. Recently, the first measurement
of the lifetime ratio $\tau({{\mathchar
28932\relax}^{0}_{b}})/\tau({{\mathchar 28931\relax}^{0}_{b}})$ was made,
yielding $\tau({{\mathchar 28932\relax}^{0}_{b}})/\tau({{\mathchar
28931\relax}^{0}_{b}})=1.006\pm 0.018\pm 0.010$ [15]. Previous ${{\mathchar
28932\relax}^{-}_{b}}$ lifetime measurements, which used ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28932\relax}^{-}$ decays, led to values of
$1.55^{+0.10}_{-0.09}\pm 0.03$ ps [16] and $1.32\pm 0.14\pm 0.02$ ps [17]. The
weighted average of these two results, along with the recent ${{\mathchar
28932\relax}^{0}_{b}}$ lifetime measurement [15], yields $\tau({{\mathchar
28932\relax}^{-}_{b}})/\tau({{\mathchar 28932\relax}^{0}_{b}})=1.00\pm 0.06$.
Improved experimental and theoretical precision of the ${{\mathchar
28932\relax}^{-}_{b}}$ lifetime will allow for a more stringent test of the
HQE prediction.

Measurements of $b$-baryon masses and isospin splittings provide information
on the interquark potential. A number of QCD-inspired models predict the
${{\mathchar 28932\relax}^{0}_{b}}$ and ${{\mathchar 28932\relax}^{-}_{b}}$
masses, or their average, which range from approximately
5780${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to
5900${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [18, 19, 20, 21, 22, 23, 24,
25, 26, 27]. More accurate predictions exist for the ${{\mathchar
28932\relax}^{-}_{b}}-{{\mathchar 28932\relax}^{0}_{b}}$ mass splitting,
estimated to be $6.24\pm 0.21$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ or
$6.4\pm 1.6$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ when extrapolating from
the measured isospin splitting $M({\mathchar 28932\relax}^{-})-M({\mathchar
28932\relax}^{0})$ or $M({{\mathchar 28932\relax}^{0}_{c}})-M({{\mathchar
28932\relax}^{+}_{c}})$, respectively [22]. The ${{\mathchar
28932\relax}^{-}_{b}}$ mass is currently known to a precision of
1.0${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [28], which is a factor of three
less precise than that of the ${{\mathchar 28932\relax}^{0}_{b}}$ baryon [15].

In this Letter, we report improved measurements of the mass and lifetime of
the ${{\mathchar 28932\relax}^{-}_{b}}$ baryon using about 1800 ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{\mathchar 28932\relax}^{0}_{c}}{{\pi}^{-}}$,
${{\mathchar 28932\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$
signal decays. The measurements are normalized using the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{-}}$,
${{\mathchar 28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ decay as
a reference. Charge conjugate processes are implied throughout.

The measurements use proton-proton ($pp$) collision data samples, collected by
the LHCb experiment, corresponding to an integrated luminosity of
3.0$\mbox{\,fb}^{-1}$, of which 1.0$\mbox{\,fb}^{-1}$ was recorded at a
center-of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$ and
2.0$\mbox{\,fb}^{-1}$ at 8$\mathrm{\,Te\kern-1.00006ptV}$. The LHCb detector
[29] 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, which provides a
momentum measurement with precision of about 0.5% from 2$-$100
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter resolution of
20${\,\upmu\rm m}$ for particles with large transverse momentum ($p_{\rm T}$).
The polarity of the dipole magnet is reversed periodically throughout data-
taking to reduce asymmetries in the detection of charged particles. Ring-
imaging Cherenkov detectors [30] are used to distinguish charged hadrons.
Photon, electron and hadron candidates are identified using a calorimeter
system, followed by detectors to identify muons [31].

The trigger [32] 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 [32, 33]. About 57% of the selected $X_{b}$ events
are triggered at the hardware level by one or more of the $X_{b}$ final-state
particles. (Throughout, we use $X_{b}$ ($X_{c}$) to refer to either a
${{\mathchar 28932\relax}^{-}_{b}}$ (${{\mathchar 28932\relax}^{0}_{c}}$) or
${{\mathchar 28931\relax}^{0}_{b}}$ (${{\mathchar 28931\relax}^{+}_{c}}$)
baryon.) The remaining 43% are triggered only on other activity in the event.
We refer to these two classes of events as triggered on signal (TOS) and
triggered independently of signal (TIS). The software trigger requires a two-,
three- or four-track secondary vertex with a large scalar $p_{\rm T}$ sum of
the particles and a significant displacement from the primary $pp$ interaction
vertices (PVs). At least one particle should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be inconsistent with coming
from any of the PVs. The signal candidates are required to pass a multivariate
software trigger selection algorithm [33].

Proton-proton collisions are simulated using Pythia [34, *Sjostrand:2007gs]
with a specific LHCb configuration [36]. Decays of hadronic particles are
described by EvtGen [37], in which final-state radiation is generated using
Photos [38]. The interaction of the generated particles with the detector and
its response are implemented using the Geant4 toolkit [39,
*Agostinelli:2002hh] as described in Ref. [41]. The $X_{c}$ final states are
modeled using a combination of resonant and nonresonant contributions to
reproduce the substructures seen in data.

Signal ${{\mathchar 28932\relax}^{-}_{b}}$ (${{\mathchar
28931\relax}^{0}_{b}}$) candidates are formed by combining in a kinematic fit
a ${{\mathchar 28932\relax}^{0}_{c}}\rightarrow
p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ (${{\mathchar 28931\relax}^{+}_{c}}\rightarrow
p{{K}^{-}}{{\pi}^{+}}$) candidate with a ${{\pi}^{-}}$ candidate (referred to
as the bachelor). The $X_{b}$ candidate is included in the fit to each PV and
is then associated with the one for which the $\chi^{2}$ increases by the
smallest amount. The kinematic fit exploits PV, $X_{b}$ and $X_{c}$ decay-
vertex constraints to improve the mass resolution. The $X_{c}$ decay products
are each required to have $\mbox{$p_{\rm
T}$}>100$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, and the bachelor pion is
required to have $\mbox{$p_{\rm
T}$}>500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. All final-state particles
from the signal candidate are required to have trajectories that are
significantly displaced from the PV and to pass particle identification (PID)
requirements. The ${{K}^{-}}$ and ${{\pi}^{+}}$ PID efficiencies are
determined from ${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$,
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ calibration samples, whereas the
proton PID efficiency is determined from simulation. The PID efficiencies are
reweighted to account for different momentum spectra and track occupancies
between the calibration and signal samples. The efficiencies of the PID
requirements on the ${{\mathchar 28932\relax}^{0}_{c}}$ and ${{\mathchar
28931\relax}^{+}_{c}}$ final states are 80% and 86%, respectively. Mass vetoes
are used to suppress cross-feeds from misidentified
$D^{+}_{(s)}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$,
$D^{*+}\rightarrow{{D}^{0}}({{K}^{+}}{{K}^{-}}){{\pi}^{+}}$, and
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decays faking
${{\mathchar 28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ decays,
as in Ref. [15]. The difference between the ${{\mathchar
28932\relax}^{0}_{c}}$ (${{\mathchar 28931\relax}^{+}_{c}}$) candidate mass
and the known value [9] is required to be less than
14${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
(20${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$), which is about 2.5 times the
mass resolution.

To improve the signal-to-background ratio, we employ a boosted decision tree
(BDT) discriminant [42, 43] built from the same variables used in Ref. [15].
To train the BDT, the kinematic distributions of the signal are modeled using
simulated decays. The background is modeled using signal candidates with
$X_{b}$ invariant mass greater than
300${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ above the signal peak mass. To
increase the size of the background sample for the ${{\mathchar
28932\relax}^{-}_{b}}$ BDT training, we also include events in the
${{\mathchar 28932\relax}^{0}_{c}}$ sideband regions,
$20<|M(p{{K}^{-}}{{K}^{-}}{{\pi}^{+}})-M({{\mathchar
28932\relax}^{0}_{c}})|<50$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The BDT
requirement is chosen to minimize the expected ${{\mathchar
28932\relax}^{-}_{b}}$ relative yield uncertainty, corresponding to a
selection efficiency of 97% (50%) for signal (combinatorial background). The
fraction of events with multiple candidates is below 1% (mostly one extra
candidate) over the full fit range in both the signal and normalization modes.
All candidates are kept.

The invariant mass signal shapes are obtained from simulated ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{\mathchar 28932\relax}^{0}_{c}}{{\pi}^{-}}$
and ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ decays. They are each modeled by the sum of
two Crystal Ball (CB) functions [44] with a common mean as

$\displaystyle f_{\rm sig}^{{{\mathchar 28931\relax}^{0}_{b}}}$
$\displaystyle=f_{\rm low}\times{\rm
CB}_{-}(m_{0},\sigma_{-},\alpha_{-},n)+(1-f_{\rm low})\times{\rm
CB}_{+}(m_{0},\sigma_{+},\alpha_{+},n)$ (1) $\displaystyle f_{\rm
sig}^{{{\mathchar 28932\relax}^{-}_{b}}}$ $\displaystyle=f_{\rm low}\times{\rm
CB}_{-}(m_{0}^{\prime},f_{\sigma}\sigma_{-},f_{\alpha_{-}}\alpha_{-},n)+(1-f_{\rm
low})\times{\rm
CB}_{+}(m_{0}^{\prime},f_{\sigma}\sigma_{+},f_{\alpha_{+}}\alpha_{+},n).$ (2)

The CB functions each include a Gaussian component to describe the core of the
mass distribution, as well as power-law tails to describe the radiative tail
below (CB-) and the non-Gaussian resolution above (CB+) the signal peak. The
extent of these tails is governed by the width and tail parameters,
$\sigma_{\pm}$ and $\alpha_{\pm}$, respectively. The parameter $m_{0}$ is the
fitted ${{\mathchar 28931\relax}^{0}_{b}}$ mass, and $m_{0}^{\prime}\equiv
m_{0}+\delta M$ is the ${{\mathchar 28932\relax}^{-}_{b}}$ mass, written in
terms of the fitted mass difference, $\delta M$, between the two signals. The
low-mass CB width, $\sigma_{-}$, is expressed in terms of the high-mass width
using $\sigma_{-}=r_{\sigma}\sigma_{+}$. The parameters $f_{\sigma}$ and
$f_{\alpha_{\pm}}$ allow for possible differences in the mass resolutions and
tail parameters, respectively, between the signal and normalization modes. We
fix the power $n=10$ and $f_{\rm low}=0.5$ to minimize the number of
correlated parameters in the signal shape. The parameters $r_{\sigma}$,
$f_{\alpha_{+}}$, $f_{\alpha_{-}}$, and $f_{\sigma}$ are determined from
simulated decays, and they are consistent with unity. These four parameters
are fixed in fits to the data to the values from simulation, while
$\sigma_{+}$, $\alpha_{+}$ and $\alpha_{-}$ are freely varied, along with
$m_{0}$ and $\delta M$.

The invariant mass spectra also include partially reconstructed $b$-baryon
background contributions, misidentified ${{K}^{-}}$ in $X_{b}\rightarrow
X_{c}{{K}^{-}}$ decays, as well as random track combinations, primarily from
false $X_{c}$ candidates. The main source of partially reconstructed
background is from $X_{b}\rightarrow X_{c}\rho^{-}$ decays, where a
${{\pi}^{0}}$ from the $\rho^{-}$ decay is not used to form the candidate. Its
shape is obtained from simulated ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}\rho^{-}$
decays, and is assumed to be the same for both the signal and normalization
modes, apart from a shift in the overall mass spectrum. A contribution from
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{\mathchar
28934\relax}_{c}^{+}{{\pi}^{-}},~{}{\mathchar
28934\relax}_{c}^{+}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{0}}$
decays is also expected to populate the ${{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ mass spectrum, and its shape is taken to be
the same to that of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}\rho^{-}$
signal. An additional contribution from partially reconstructed ${\mathchar
28932\relax}_{b}$ decays is found, through a study of the ${{\mathchar
28931\relax}^{+}_{c}}$ sidebands, to populate the ${{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ mass spectrum. This background is modeled
through a fit to the ${{\mathchar 28931\relax}^{0}_{b}}$ candidate mass
spectrum obtained using the lower and upper ${{\mathchar
28931\relax}^{+}_{c}}$ mass sidebands. The shape of the background from
misidentified $X_{b}\rightarrow X_{c}{{K}^{-}}$ decays is taken from
simulation. The misidentification rate of 3.1% is obtained from
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ calibration samples, reweighted in
$p_{\rm T}$, $\eta$ and number of tracks to match the distributions observed
in data. No peaking contributions from charmless backgrounds are observed when
studying the $X_{b}$ mass spectra using the $X_{c}$ mass sidebands. The
combinatorial background is modeled using an exponential function with a
freely varying slope.

The ${{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{-}}$ and ${{\mathchar
28932\relax}^{0}_{c}}{{\pi}^{-}}$ mass spectra are fit simultaneously using a
binned maximum likelihood fit. The results of the fit are shown in Fig. 1. A
total of $1799\pm 46$ ${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{c}}{{\pi}^{-}}$ and $(220.0\pm 0.5)\times 10^{3}$
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ signal decays are observed. The mass
difference is measured to be

$\displaystyle\delta M\equiv M({{\mathchar
28932\relax}^{-}_{b}})-M({{\mathchar 28931\relax}^{0}_{b}})$
$\displaystyle=178.36\pm 0.46{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$

where the uncertainty is statistical only.

Figure 1: Invariant mass spectrum, along with the fit projections, for (left)
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ and (right) ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{\mathchar 28932\relax}^{0}_{c}}{{\pi}^{-}}$
candidates.

The observed signals are also used to measure the ${{\mathchar
28932\relax}^{-}_{b}}$ baryon lifetime relative to that of the ${{\mathchar
28931\relax}^{0}_{b}}$ baryon. We measure the efficiency-corrected yields in
six bins of measured decay time, as given in Table 1. The ratio of efficiency-
corrected yields depends exponentially on decay time as $N_{\rm
cor}[{{{\mathchar 28932\relax}^{-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{c}}{{\pi}^{-}}}](t)/N_{\rm cor}[{{{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}}](t)=e^{\beta t}$, where
$\beta=1/\tau({{\mathchar 28931\relax}^{0}_{b}})-1/\tau({{\mathchar
28932\relax}^{-}_{b}})$. Many systematic uncertainties cancel to first order
in the ratio, such as those associated with the time resolutions and relative
acceptances.

The yields in each time bin are obtained using the results from the full fit
with the signal shape parameters fixed. No dependence of the signal shapes on
decay time is observed in simulated decays, as expected. The background shape
parameters are also fixed, except for the combinatorial background shape
parameter, and one of the $X_{b}\rightarrow X_{c}\rho$ shape parameters, which
is seen to have a dependence on decay time. The signal yields in each of the
time bins are shown in Table 1.

Table 1: Fitted yields of ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{-}}$ and ${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{{\mathchar 28932\relax}^{0}_{c}}{{\pi}^{-}}$ in each time bin. Uncertainties are statistical only. Decay time (ps) | ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{-}}$ | ${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{{\mathchar 28932\relax}^{0}_{c}}{{\pi}^{-}}$
---|---|---
$0-1$ | $38,989\pm 212$ | $260\pm 17$
$1-2$ | $79,402\pm 299$ | $629\pm 27$
$2-3$ | $48,979\pm 233$ | $436\pm 22$
$3-4$ | $26,010\pm 169$ | $232\pm 16$
$4-6$ | $19,651\pm 147$ | $177\pm 14$
$6-9$ | $5794\pm 79$ | $69\pm 9$

The relative acceptance, shown in Fig. 2, is obtained using simulated decays
after applying all event selection criteria.

Figure 2: Ratio of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{-}}$
to the ${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{c}}{{\pi}^{-}}$ selection efficiencies as a function of
decay time. The uncertainties are due to the finite size of the simulated
samples.

The efficiency for reconstructing the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{\mathchar 28932\relax}^{0}_{c}}{{\pi}^{-}}$
mode is about a factor of two lower than that of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{-}}$
decay due to the extra particle in the final state and the lower average
momentum of the final-state particles. The relative efficiency,
$\epsilon({{\mathchar 28931\relax}^{0}_{b}})/\epsilon({{\mathchar
28932\relax}^{-}_{b}})$, is nearly uniform, with a gradual increase for decay
times below 2 ps. This increase is expected, because the ${{\mathchar
28931\relax}^{+}_{c}}$ lifetime is about twice that of the ${{\mathchar
28932\relax}^{0}_{c}}$ baryon, and the correspondingly larger impact
parameters are favored by the software trigger and offline selections, most
notably when the $X_{b}$ decay time is small.

The ratios of corrected yields and the exponential fit are shown in Fig. 3.
The points are displayed at the average time value in the bin assuming an
exponential time distribution with mean 1.54 ps, which is the average of the
known ${{\mathchar 28931\relax}^{0}_{b}}$ and fitted ${{\mathchar
28932\relax}^{-}_{b}}$ lifetimes. Choosing either the ${{\mathchar
28931\relax}^{0}_{b}}$ or the fitted ${{\mathchar 28932\relax}^{-}_{b}}$
lifetime leads to a negligible change in the result. The fitted value is
$\beta=0.0557\pm 0.0160$ ps-1, where the uncertainty is statistical only.
Using $\tau({{\mathchar 28931\relax}^{0}_{b}})=1.468\pm 0.009\pm 0.008$ ps
[45], we find

$\displaystyle r_{\tau}\equiv\frac{\tau_{{{\mathchar
28932\relax}^{-}_{b}}}}{\tau_{{{\mathchar 28931\relax}^{0}_{b}}}}$
$\displaystyle=1.089\pm 0.026\,({\rm stat}).$

Figure 3: Corrected yield ratio, $N_{\rm cor}({{\mathchar
28932\relax}^{-}_{b}})/N_{\rm cor}({{\mathchar 28931\relax}^{0}_{b}})$ in bins
of decay time, along with the exponential fit. The uncertainties are
statistical only.

Several consistency checks are performed, including comparing the mass
differences obtained from 7$\mathrm{\,Te\kern-1.00006ptV}$ versus
8$\mathrm{\,Te\kern-1.00006ptV}$ data, opposite magnet polarities, $X_{b}$
versus $\overline{X}_{b}$ samples, and different trigger selections. In all
cases, the results are consistent with statistical fluctuations of independent
samples. In addition, the analysis is carried out using 15,500
${{{B}^{-}}}\rightarrow{{D}^{0}}{{\pi}^{-}}$,
${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ signal decays
for normalization. The ${{\mathchar 28932\relax}^{-}_{b}}$ mass and lifetime
results agree with the above values to better than one standard deviation,
considering only the uncertainty due to the ${{\mathchar
28931\relax}^{0}_{b}}$ and ${{{B}^{-}}}$ masses and lifetimes.

The measurements of $M({{\mathchar 28932\relax}^{-}_{b}})$ and
$\tau({{\mathchar 28932\relax}^{-}_{b}})$ are subject to systematic
uncertainties, but the largest contributions cancel to first order in $\delta
M$ and $r_{\tau}$. For the mass difference measurement, the effect of the
momentum scale uncertainty of 0.03% [46] is investigated by shifting the
momenta of all final-state particles in simulated decays by this amount,
leading to an uncertainty on $\delta M$ of
0.08${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Because the signal mode has
one more particle than the normalization mode, the correction for energy loss
in the detector material leads to an additional uncertainty of
0.06${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [46]. Uncertainty due to the
signal modeling is 0.06${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, obtained by
shifting all fixed parameters by their uncertainties, and adding the shifts in
$\delta M$ from the nominal value in quadrature. For the background model,
several variations from the nominal fit are investigated, including (a) using
a second-order polynomial to describe the combinatorial background, (b)
allowing the fixed parameters in the partially reconstructed background to
vary, (c) removing the ${\mathchar 28932\relax}_{b}$ background component, (d)
a 20% relative increase in the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{\mathchar 28932\relax}^{0}_{c}}{{K}^{-}}$
cross-feed, and (e) varying the fit range. The changes in $\delta M$ are added
in quadrature to obtain the background uncertainty of
0.11${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Adding all sources of
uncertainty in quadrature leads to a systematic uncertainty in $\delta M$ of
0.16${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The largest source of systematic uncertainty in $r_{\tau}$ is the limited size
of the simulated samples, which contributes an uncertainty of 0.010. The
simulated efficiencies are averaged over TOS and TIS events in the simulation,
of which the former comprises 67% of the sample, compared to 57% in data.
While the values of $r_{\tau}$ are statistically compatible between these two
samples, if the efficiencies from simulation are reweighted to match the
composition observed in data, a change in $r_{\tau}$ of 0.004 is found. This
shift is assigned as a systematic uncertainty. Variation in the signal and
background models lead to a negligible change in $r_{\tau}$. We also consider
possible different performances of the BDT in data versus simulation by
correcting the data with an efficiency obtained with a tighter BDT
requirement. The difference of 0.001 is assigned as a systematic uncertainty.
For the proton efficiency, we use the values obtained from simulation. By
varying the proton PID requirements, a maximal change of 0.001 is found, which
is assigned as a systematic uncertainty. To investigate possible effects due
to the larger ${{\mathchar 28931\relax}^{+}_{c}}$ lifetime (than the
${{\mathchar 28932\relax}^{0}_{c}}$), we reject candidates with $ct$ larger
than 150 $\mu$m. The difference of 0.003 from the nominal result is assigned
as a systematic uncertainty. In total, the systematic uncertainty on
$r_{\tau}$ is 0.011.

In summary, we use a $pp$ collision data sample corresponding to
3.0$\mbox{\,fb}^{-1}$ of integrated luminosity to improve the precision of the
${{\mathchar 28932\relax}^{-}_{b}}$ mass and lifetime by a factor of two over
the previous best measurements. The resulting mass difference and relative
lifetime are

$\displaystyle M({{\mathchar 28932\relax}^{-}_{b}})-M({{\mathchar
28931\relax}^{0}_{b}})$ $\displaystyle=178.36\pm 0.46\pm
0.16{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$
$\displaystyle\frac{\tau_{{{\mathchar
28932\relax}^{-}_{b}}}}{\tau_{{{\mathchar 28931\relax}^{0}_{b}}}}$
$\displaystyle=1.089\pm 0.026\pm 0.011,$

where the uncertainties are statistical and systematic, respectively. Using
the measured ${{\mathchar 28931\relax}^{0}_{b}}$ mass [47] and lifetime [45],
we find

$\displaystyle M({{\mathchar 28932\relax}^{-}_{b}})$ $\displaystyle=5797.72\pm
0.46\pm 0.16\pm 0.26_{{{\mathchar
28931\relax}^{0}_{b}}}~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$
$\displaystyle\tau_{{{\mathchar 28932\relax}^{-}_{b}}}$
$\displaystyle=1.599\pm 0.041\pm 0.018\pm 0.012_{{{\mathchar
28931\relax}^{0}_{b}}}~{}{\rm ps},$

where the last uncertainty is due to the precision on the ${{\mathchar
28931\relax}^{0}_{b}}$ lifetime. Using the measurements of the ${{\mathchar
28932\relax}^{0}_{b}}$ mass difference and relative lifetime, $M({{\mathchar
28932\relax}^{0}_{b}})-M({{\mathchar 28931\relax}^{0}_{b}})=172.44\pm
0.39\,\pm 0.17$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$\tau_{{{\mathchar 28932\relax}^{0}_{b}}}/\tau_{{{\mathchar
28931\relax}^{0}_{b}}}=1.006\pm 0.018\pm 0.010$ [15], we obtain

$\displaystyle M({{\mathchar 28932\relax}^{-}_{b}})-M({{\mathchar
28932\relax}^{0}_{b}})$ $\displaystyle=5.92\pm 0.60\pm
0.23{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
$\displaystyle\frac{\tau_{{{\mathchar
28932\relax}^{-}_{b}}}}{\tau_{{{\mathchar 28932\relax}^{0}_{b}}}}$
$\displaystyle=1.083\pm 0.032\pm 0.016.$

The measured isospin splitting between the ${{\mathchar 28932\relax}^{-}_{b}}$
and ${{\mathchar 28932\relax}^{0}_{b}}$ baryons is consistent with the
prediction in Ref. [22] of $6.24\pm
0.21$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The relative lifetime is 2.3
standard deviations larger than unity, giving a first indication that the
${{\mathchar 28932\relax}^{-}_{b}}$ baryon lifetime is larger than that of the
${{\mathchar 28932\relax}^{0}_{b}}$ baryon. This result is consistent with the
theoretical expectations of $\tau_{{{\mathchar
28932\relax}^{-}_{b}}}/\tau_{{{\mathchar 28932\relax}^{0}_{b}}}=1.05\pm 0.07$
[12] and $\tau_{{{\mathchar 28932\relax}^{-}_{b}}}/\tau_{{{\mathchar
28931\relax}^{0}_{b}}}=1.19^{+0.07}_{-0.06}$ [14], based on the HQE.

## Acknowledgements

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, HGF and MPG (Germany); SFI (Ireland);
INFN (Italy); FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA
(Romania); MinES and FANO (Russia); MinECo (Spain); SNSF and SER
(Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA). The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages on which we depend. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia). Individual groups or members have received support from EPLANET,
Marie Skłodowska-Curie Actions and ERC (European Union), Conseil général de
Haute-Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR
(Russia), XuntaGal and GENCAT (Spain), Royal Society and Royal Commission for
the Exhibition of 1851 (United Kingdom).

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van Tilburg41, V. Tisserand4, M. Tobin39, J. Todd57, S. Tolk42, L.
Tomassetti16,f, D. Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4,
S. Tourneur39, M.T. Tran39, M. Tresch40, A. Trisovic38, A. Tsaregorodtsev6, P.
Tsopelas41, N. Tuning41, M. Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin64, U.
Uwer11, C. Vacca15, V. Vagnoni14, G. Valenti14, A. Vallier7, R. Vazquez
Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S. Vecchi16, J.J.
Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, B. Viaud7, D.
Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,p, A. Vollhardt40, D.
Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß63, J.A. de
Vries41, R. Waldi63, C. Wallace48, R. Wallace12, J. Walsh23, S. Wandernoth11,
J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M. Whitehead48, J.
Wicht38, D. Wiedner11, G. Wilkinson55,38, M.P. Williams45, M. Williams56, H.W.
Wilschut66, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, K. Wyllie38, Y. Xie61, Z.
Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14, M.
Zavertyaev10,b, L. Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A.
Zhokhov31, L. Zhong3.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
63Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
64National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
65Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
66Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 41
67Celal Bayar University, Manisa, Turkey, associated to 38

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
vPolitecnico di Milano, Milano, Italy


# The determination of $\alpha$-spectroscopic factors and ANC of 16O states
using 12C(20Ne,16O)16O reaction at $E_{lab}$=150 MeV incident energy

Ashok Kumar Mondal1 C. Basu1<EMAIL_ADDRESS>S. Adhikari2 C.
Bhattacharya3 T. K. Rana3 S. Kundu3 S. Manna3 R. Pandey3 P. Roy3 A. Sen3 J. K.
Meena3 A. K. Saha3 J. K. Sahoo3 Dipali Basak1 Tanmay Bar1 Haridas Pai1 A.
Bisoi4 A. K. Mitra1 Piyasi Biswas1 1Nuclear Physics Division, Saha Institute
Of Nuclear Physics, HBNI, 1/AF Bidhan Nagar, Kolkata- 700064, INDIA.  2Techno
India University, EM-4/1, Sector-V, Salt Lake, Kolkata-700091, INDIA.
3Variable Energy Cyclotron Centre, Kolkata - 700064, INDIA.  4Indian Institute
of Engineering Science and Technology, Shibpur, Howrah -711103, INDIA.

###### Abstract

The 12C(20Ne,16O)16O $\alpha$-transfer reaction at $E_{lab}$=150 MeV is first
time used to determine the ANC of the 6.92 MeV and 7.12 MeV states of 16O. The
20Ne+12C potential parameters are also obtained from elastic scattering. The
direct reaction code FRESCO is used to determine the $\alpha$-spectroscopy
factor ($S_{\alpha}$) of the three states of 16O (6.92 MeV, 7.12 MeV and 11.52
MeV) and ANC of the two states (6.92 MeV and 7.12 MeV) of 16O. The extracted
ANC and $S_{\alpha}$ are compareable to previous measurements.

###### pacs:

Valid PACS appear here

##  I Introduction

The Asymptotic Normalization Co-efficient (ANC) method [1,2] is an indirect
method to study astrophysical reactions at low energies. Such reactions take
place in stars at relative energies much below the Coulomb barrier of the two
interacting nuclei (except in neutron induced reactions). As a result, the
cross-sections are very small and their direct measurement with reasonable
accuracy is very difficult or almost impossible with presently available
techniques. R-matrix extrapolation [3] of the cross-section measured at higher
energy to the Gamow energy is the solution of this problem. Additionally, if
the main component of the capture cross-section is external capture then the
determination of the ANC can evalute the S-factor at zero energy. The
12C($\alpha$,$\gamma$) reaction is such a reaction for which the ANC method is
most suitable [4]. The rate of this reaction greatly affects the resulting
ratio of 12C to 16O and also the final fate of the star (i.e., black hole or
neutron star). At 300 keV (Gamow energy) the cross-section of this reaction is
of the order of $10^{-17}$ barn and its direct measurement is almost
impossible. The only method to determine the 12C($\alpha$,$\gamma$)16O
reaction cross-section at 300 keV is by R-matrix extrapolation . The
extrapolation relies on the $\alpha$-spectroscopic properties of 16O states
[5,6,7,8,9,10]. One can allow these parameters to vary freely in the
extrapolation process. But a more physical away is to determine these
parameters from some indirect method such as the ANC technique. The alpha
capture in the 12C($\alpha$,$\gamma$)16O reaction is an external capture
process and proceeds through mainly two sub-threshold states (6.92 MeV (2+)
and 7.12 MeV (1-)) of 16O. The ANC of these two sub-threshold states play a
crucial role in determination of the capture cross-section. The 12C(6Li,d) and
12C(7Li,t) $\alpha$-transfer reactions [2, 5, 7, 11, 12] have been mainly used
to determine the ANC at both below and above Coulomb barrier energies.
Besides, 12C(11B,7Li)16O reaction has been also used for the same purpose
[13]. Though, at sub-Coulomb energeis the determination of ANC is more model
independent, the above barrier measurements are more convenient due to larger
cross-sections.

In this work, a new reaction 12C(20Ne,16O)16O is used for the first time to
determine the $S_{\alpha}$ and ANC of 16O states and using this ANC of 16O
states we have extracted $S_{E2}$ value at 300 keV by R-matrix extrapolation.
The measurements are carried out at an incident energy of 150 MeV.

##  II Experimental Details

The experiment was carried out at the K130 room temperature Cyclotron centre
at VECC, Kolkata. The 1 m scattering chamber at channel 2 was used for the
measurements. Three $\Delta$E-E telescopes were setup inside the chamber. One
telescope consisted of $\Delta$E-E setup with (50 mm x 50 mm) silicon strip
detectors. The $\Delta$E strip detector was 50 $\mu$m thick and the E strip (
double sided) 500 $\mu$m. The other telescopes (T1 and T2) were 15 $\mu$m-
1000 $\mu$m surface barrier detectors. The target was self supporting 660
$\mu$gm/$cm^{2}$ 12C foils. A gold target was mounted in the ladder for energy
calibration purpose. The beam current was stable at around 10-20 pnA.There
were 16 strips in the $\Delta$E detector with 100 $\mu$m dead layer between
adjacent strips. The centre to centre distance between adjacent strip was 3
mm. The strip telescope covered angles from 14.82o to 34.18o whereas the low
threshold surface barrier telescopes were used to measure the cross-sections
at backward angles from 30o to 52o. Typical energy resolutions of the
$\Delta$E and E strip detectors were 40 keV and 60 keV respectively. The solid
angles subtended by the telescopes at the target centre were 0.796 msr(strip
detector),0.128 msr(T1 telescope) and 0.2716 msr (T2 telescope) respectively.
The 20Ne + 12C reaction have been measured in the angular range ${15}^{o}$ to
${52}^{o}$ at $E_{lab}$=150 MeV energy. The distance between target and strip
detector $\Delta$E-E telescopes was 184.15 mm and with surface barrier
telescopes were 235 mm. The dimension of the collimator in the strip detector
was 50 mm x 9 mm (rectangular) and for T1 and T2, 3 mm diameter circular and 3
mm x 5 mm rectangular respectively. VME data aquisition system was used in the
experiment.

##  III Results and Discussion

In this experiment ${}^{20}{Ne}^{10^{+}}$ beam of energy $E_{lab}$=150 MeV was
incident on a 12C target and the 2-D spectrum for a representative case of
19.82o for the strip telescope is shown in fig.1 and that for the telescope T1
at 36o is shown in fig.2.

Figure 1: 2-D energy spectrum of the strip detector $\Delta$E-E telescope for
the 20Ne+12C system at $\Theta_{lab}$=19.82o for E${}^{lab}_{{}^{20}Ne}$=150
MeV. Figure 2: 2-D energy spectrum of the SB detector $\Delta$E-E telescope T1
for the 20Ne+12C system at $\Theta_{lab}$=36o for E${}^{lab}_{{}^{20}Ne}$=150
MeV. Figure 3:  Energy spectrum of elastic scattering of 20Ne from 12C at
$\Theta_{lab}$=19.82o for E${}^{lab}_{{}^{20}Ne}$=150 MeV using strip detector
$\Delta$E-E telescope. Here at higher energies there is the strong elastic
peak of 20Ne and at lower energies there are 20Ne nonelastic fragments.

(a) $\Theta_{lab}$=19.82o

(b) $\Theta_{lab}$=20.77o

(c) $\Theta_{lab}$=21.73o

(d) $\Theta_{lab}$=22.68o

(e) $\Theta_{lab}$=23.64o

Figure 4:  Energy spectrum of the 16O in strip detector $\Delta$E-E telescope
for the 12C(20Ne,16O)16O reaction at different angles for
E${}^{lab}_{{}^{20}Ne}$=150 MeV which are fitted by moving source model [14].
The red line fit the low energy continuous background using moving source
model.

Emitted fragments (2$\geq$Z$\leq$10) are clearly seperated as seen in these
plots. Fig.3 shows the projected elastic and nonelastic fragments at the angle
$\Theta_{lab}$=19.82o. Gating on the Z=8 curve from the 2-D spectrum in fig.1
and projecting on the energy axis, we obtain the energy spectrum of the
emitted 16O fragments. These spectra at various angles are shown in fig.4. The
spectrum has a broad continuous low energy section and some discrete peaks at
higher energy. These discrete peaks can results from transfer reactions or
compound emissions populating discrete low lying states of 16O. The three
discrete peaks were identified at 6.92 MeV, 7.12 MeV and 11.52 MeV. In order
to obtain the cross-sections of these three states, the cross-section of the
continuous part has to be estimated as they contribute to the background of
the discrete peaks.

The shape of the continuous part suggest a statistical process. In order to
calculate the cross-section for emission of such heavy fragments (Z=8) from
statistical process, we used the moving source model [14]. The parameters in
this model are $N_{0}$, Z$E_{C}$, $E_{1}$, and T, Where $N_{0}$ is an overall
normalization constant, $E_{C}$ is the kinetic energy gained by the light
particle of charge Z due to the Coulomb repulsion from the target, $E_{1}$ is
the kinetic energy of a particle at rest in the moving frame and T is the
source temperature. The value of these parameters required to calculate the
16O spectrum at various angles are given in the Table I. The cross-sections of
the discrete peaks are estimated by subtracting the background calculated from
moving source model and are plotted in fig.7.

(a) $\Theta_{lab}$=19.82o

(b) $\Theta_{lab}$=20.77o

(c) $\Theta_{lab}$=21.73o

(d) $\Theta_{lab}$=22.68o

(e) $\Theta_{lab}$=23.64o

Figure 5:  Fitting of the 6.92 MeV and 7.12 MeV peaks of the 16O after substracting the low energy continuous background by moving source model [14] at different angles. Table 1: The parameters in moving source model at various angles are given below. Lab angle | $N_{0}$ |  Z$E_{C}$ (MeV) |  $E_{1}$ (MeV) |  T (MeV)
---|---|---|---|---
19.82 |  239786 $\pm$ 86754 |  36.15 $\pm$ 0.173 | 45 |  0.677 $\pm$ 0.0301
20.77 |  4581 $\pm$ 833 |  46.19 $\pm$ 0.166 | 28 |  0.91 $\pm$ 0.0423
21.73 |  3322 $\pm$ 54 |  53.25 $\pm$ 0.126 | 20 |  0.9 $\pm$ 0.035
22.68 |  2025 $\pm$ 30 |  58.67 $\pm$ 0.0912 | 13 |  0.85 $\pm$ 0.032
23.64 |  696 $\pm$ 9 |  67.19 $\pm$ 0.0.036 |  1.89 $\pm$ 0.56 |  1.45 $\pm$ 0.052

Indirect method uses transfer reactions to determine the ANC of the bound
states of the nuclei of interest. In this work, the transfer reaction (
12C(20Ne,16O)16O) is chosen to produce 16O through $\alpha$-transfer, the ANC
of whose states have to be determined. The measured cross-sections are then
compared to a single particle transfer reaction model such as the DWBA
(Distorted Wave Born Approximation)[1] theory. The $\alpha$-spectroscopic
factor ($S_{\alpha}$) of a nuclear state can be extracted from
$\alpha$-transfer reaction by a normalization of the experimental data with
the theoretical cross-section. Thus,

$(\frac{d\sigma}{d\Omega})_{Expt}=S_{1}S_{\alpha}(\frac{d\sigma}{d\Omega})_{Theo}$
(1)

where $S_{1}$ is the spectroscopic factor for the $\alpha$ \+ 16O
configuration of the 20Ne ground state and $S_{\alpha}$ is the $\alpha$ \+ 12C
spectroscopic factor for a state of 16O. The square of the ANC (C2) of a
particular state is related to the alpha spectroscopic factor ($S_{\alpha}$)
via the single particle ANC b2 as

$C^{2}=S_{\alpha}b^{2}$ (2)

The single particle ANC b is the normalization of the bound state wave
function of 16O at large radii with respect to the Whittaker function and is
calculated from a suitable binding potential.

In fig.7, the measured 16O angular distribution (symbol) is shown for the 6.92
MeV, 7.12 MeV and 11.52 MeV states. The calculated cross-sections (solid
lines) are obtained by using the code FRESCO [15] (version fres2.9) in the
framework of the DWBA theory. The required exit channel (16O + 16O) potential
and the core-core 16O + 12C potential are obtained respectively from [6] and
[16]. The entrance channel potential are extracted from elastic scattering
data measured in the present work. The $\alpha$ \+ 16O and $\alpha$ \+ 12C
binding real potentials have a Gaussian form factor. The depth of the
potential is adjusted to fit the respective seperation energies of $\alpha$ in
20Ne and 16O. The optical model analysis for the elastic scattering data is
performed with a Woods-Saxon (WS) potential i.e

$U(r)=\frac{-V_{o}}{1+exp(\frac{r-R_{o}}{a_{o}})}+\frac{-iW}{1+exp(\frac{r-R_{w}}{a_{w}})}$

where $V_{o}$, $R_{o}$ and $a_{o}$ are real potential parameters and W,
$R_{w}$ and $a_{w}$ are imaginary potential parameters (Where $R_{c}$=
1.2$A_{T}^{1/3}$).
Elastic scattering of 20Ne on 12C at $E_{lab}$=150 MeV from $40^{o}$ to
$65^{o}$ in the C.M system is used to extract the potential parameters for the
20Ne+12C system. The open black square in fig.6 are the experimental elastic
scattering data measured in the present work and dotted red line is the
SFRESCO fit [15]. The extracted potential parameters are shown in Table.2.
This parameters are used in $S_{\alpha}$ and the calculation of
12C(20Ne,16O)16O transfer cross-section for the ANC of 16O states.

Figure 6: Angular distribution of 12C(20Ne,20Ne)12C elastic scattering cross-
section at $E_{l}ab$=150 MeV from $40^{o}$ to $65^{o}$ in C.M system. Here
open black square are experimental elastic scattering cross-section data with
error and dotted red line is the SFRESCO calculation.

(a) 6.92 MeV state of 16O

(b) 7.12 MeV state 16O

(c) 11.52 MeV state 16O

Figure 7: Comparison of DWBA calculations using FRESCO of 12C(20Ne,16O)16O
$\alpha$-transfer angular distribution of 6.92 MeV, 7.12 MeV and 11.52 MeV
states of 16O with experimental data. The black square are experimental data
with error and solid red line is the FRESCO calculation and solid blue line is
the FRESCO calculation after normalisation to the experimental data for (a)
6.92 MeV, (b) 7.12 MeV and (c) 11.52 MeV states of 16O respectively.

The $S_{\alpha}$ of the 6.92 MeV, 7.12 MeV and 11.52 MeV states are extracted
by a normalization of the experimental angular distributions (in Fig.7) in
terms of the calculated values using equation (1). The spectroscopic factor of
20Ne is adopted as 0.229 from [18]. The potential parameters used in the
FRESCO calculations are shown in Table II. and Table III. The comparision of
the calculations with data yields the $S_{\alpha}$ and the ANC values which
are shown in Table IV. and Table V.

Error analysis of the extracted ANC and spectroscopic factors have been
carried out. The sources of error in the spectroscopic factor are from
experimental and theoretical DWBA cross-sections. The errors in the
experimental cross-section are estimated from statistical and systematic
errors. The statistical error is about 10$\%$. The source of systematic errors
are from beam energy fluctuation, target thickness uncertainty, beam current
fluctuation, solid angle uncertainty etc. and is estimated to be 5$\%$. The
errors (statistical and systematic) are added in quadrature assuming them to
be independenet errors to obtain the error in the spectroscopic factor. The
source of the theoretical errors in the DWBA calculation are from the
uncertainty in the entrance, exit and core-core potentials. The uncertainty in
the ANC depends on the error in the spectroscopic factor and that in the
single particle ANC (b). The two errors are added in quadrature to obtain the
error in the extracted ANC value. The error in single particle ANC (b) is
about 14$\%$. Fig.8 shows the $S_{E2}$ curve with the 12C($\alpha$,$\gamma$)
E2 capture data [23,24] have been described by a four level R-matrix fit (6.92
MeV sub-threshold state, 9.85 MeV state, 11.52 MeV state and a higher
background equivalent state) assuming a radius $R_{0}$= 6.5 fm for the inner
space using the AZURE2 code [25]. The fitting is carried out with all
parameters fixed except the energy and width of the background state which are
considered to be freely varying. The ANC of the 6.92 MeV state was fixed at
(1.034 $\pm$ 0.4)$\times$ $10^{5}$ $fm^{-1/2}$ which is determined through the
12C(20Ne,16O)16O reaction in the present case. Its energy and
$\Gamma_{\gamma}$ have fixed to the values $E_{r}$= 0.2448 MeV and
$\Gamma_{\gamma}$= 97 meV [26]. For the resonance parameters of the $E_{x}$=
9.85 MeV and $E_{x}$= 11.52 MeV states, we have used the value given in [27].
All the resonance parameters used in the R-matrix fit for the astrophysical S
factor of the E2 component are given in the Table VI. The extracted $S_{E2}$
value from fig.8 at 300 keV is 52 $\pm$ 25 keV b. Comparision of astrophysical
S factors at 300 keV obtained in various experiments including this work for
E2 components are given in the Table VII.

Table 2: The potential parameters obtained from the elastic 20Ne + 12C
scattering at $E_{lab}$=150 MeV in optical model analysis.
System | Vo (MeV) | ro (fm) | ao (fm) | W (MeV) | rw (fm) | aw (fm) | Ref.
---|---|---|---|---|---|---|---
20Ne + 12C (WS) | 85.00 | 1.383 | 0.63 | 10.65 | 1.5 | 0.56 | [This work]
Table 3: The potential parameters required in the calculation are shown. System |  Vo (MeV) |  ro (fm) |  ao (fm) | W (MeV) |  rw (fm) |  aw (fm) |  Ro (fm) |  rc (fm) | Ref.
---|---|---|---|---|---|---|---|---|---
20Ne + 12C (WS) | 85.00 | 1.383 | 0.63 | 10.65 | 1.5 | 0.56 | - | 1.25 | [This work]
16O + 16O (WS) | 135.0 | 1.73 | 0.835 | 27.0 | 2.3 | 0.6 | - | 1.25 | [6]
16O + 12C (WS) | 282.2 | 0.586 | 0.978 | 13.86 | 1.183 | 0.656 | - | - | [16]
$\alpha$ \+ 12C (WS) | 85.9 | 0.916 | 2.7 | - | - | - | - | - | [17]
$\alpha$ \+ 16O (WS) | 13.496 | 2.303 | 5.4 | - | - | - | - | - | [-]
Table 4: Comparison of spectroscopic factors of the 16O states deduced from our calculation with earlier works. State of 16O (MeV) | Present work | S. Adhikari [5] | Oulebsir [19] | Bellhout [20] | Bechetti | Keely [7] | Cobern [8]
---|---|---|---|---|---|---|---
6.92 |  0.786 $\pm$ 0.56 |  0.32 $\pm$ 0.07 |  0.15 $\pm$ 0.05 |  0.37 $\pm$ 0.11 | 1.35 [9] and 4.134 [10] | 0.68 | 1.10
7.12 |  0.0196 $\pm$ 0.017 |  0.22 $\pm$ 0.07 |  0.07 $\pm$ 0.03 |  0.11 $\pm$ 0.03 |  $0.08^{+0.04}_{-0.06}$ [10] | - | 0.2
11.52 |  5.89 $\pm$ 1.56 | - | - | - | - | - | -
Table 5: Comparison of ANC of 6.92 MeV and 7.12 MeV states of 16O ontained from present work with earlier works. Experiment |  $C^{2}$ ($2^{+}$) ($10^{10}$ $fm^{-1}$) |  $C^{2}$ ($1^{-}$) ($10^{28}$ $fm^{-1}$)
---|---|---
This work |  ( 1.07 $\pm$ 0.16 ) |  ( 6.56 $\pm$ 1.45 )
A. Mondal [21,22] | 1.14 | 76.8
S. Adhikari [5] |  (3.0 $\pm$ 0.122) |  ( 303.8 $\pm$ 22.09 )
Brune [2] |  (1.29 $\pm$ 0.23) |  ( 4.33 $\pm$ 0.84 )
Avila [12] |  (1.48 $\pm$ 0.16) |  ( 4.39 $\pm$ 0.59 )
Belhout [20] |  ($1.96^{+1.41}_{-1.27}$) |  ( 3.48 $\pm$ 2.0 )
Oulebsir [19] |  (2.07 $\pm$ 0.8) |  ( 4.0 $\pm$ 1.38 )
Figure 8: Astrophysical S factor for the 12C($\alpha$,$\gamma$)16O reaction with R-matrix calculations of the E2 component. The solid square and solid star points are the experimental data from [23] and [24] respectively. The solid red line is the R-matrix fit using ANC value of the 6.92 MeV state determined in the present work.  Table 6: Resonance parameters used in the R-matrix fit of the astrophysical S factor of the E2 component. The values in the brackets are the fixed resonance parameters. $J^{\pi}$ |  $E_{x}$ (MeV) |  $E_{r}$ (MeV) |  ANC (C) in $fm^{-1/2}$ or $\Gamma_{\alpha}$ (keV) |  $\Gamma_{\gamma}$ (keV)
---|---|---|---|---
$2^{+}$ | 6.92 | [-0.244] |  C=[(1.228 $\pm$ 0.6)$\times$ $10^{5}$] |  [9.7$\times$ $10^{-5}$]
$2^{+}$ | 9.85 | [2.683] |  $\Gamma_{\alpha}$=[0.76] |  [5.7$\times$ $10^{-6}$]
$2^{+}$ | 11.52 | [4.339] |  $\Gamma_{\alpha}$=[83.0] |  [6.1$\times$ $10^{-4}$]
$2^{+}$ | Background | 7.84 |  $\Gamma_{\alpha}$= 0.257 |  1.3$\times$ $10^{-4}$
Table 7: Comparision of astrophysical S factors at 300 keV obtained in various experiments including this work for E2 components. S factor | This Work | Oulebsir[19] | Brune[2] | Tischauser[27] | Hammer[28] | Kunz[23] | NACRE[29] | Ouellet[24]
---|---|---|---|---|---|---|---|---
(0.3 MeV) |  |  |  |  |  |  |  |
$S_{E2}$ (keV b) |  52 $\pm$ 25 |  50 $\pm$ 19 | $44^{+19}_{-23}$ |  53 $\pm$ 13 |  81 $\pm$ 22 |  85 $\pm$ 30 |  120 $\pm$ 60 |  36 $\pm$ 6

##  IV Conclusions

In this experiment we have measured 12C(20Ne,16O)16O angular distribution for
the population of $\alpha$-transfer states at $E_{lab}$=150 MeV. The ANC of
the 6.92 MeV and 7.12 MeV states of 16O are determined for the first time
using this reaction. The extraction of ANC were done by using the direct
reaction code FRESCO. The required entrance channel potential of 20Ne+12C was
determined from the elastic scattering data measured in the same experiment.
We have extracted $S_{E2}$ value at 300 keV by R-matrix calculation using ANC
value of 6.92 MeV state which is determined from this reaction.

##  V Acknowledgement

The authors acknowledge the help and support of all VECC Cyclotron staff,
Jonaki Panja and Taniya Basu for the experiment.

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# Limits on Primordial Black Holes from M87

Joseph Silk
Institut d’Astrophysique de Paris, UMR7095:CNRS
UPMC-Sorbonne University, F-75014, Paris, France

Leo Stodolsky
Max-Planck-Institut für Physik (Werner-Heisenberg-Institut)
Föhringer Ring 6, 80805 München, Germany

###### Abstract

Primordial black holes in the solar mass range are a possibly significant
component of dark matter. We show how an argument relating the deflection of
light by such black holes in the density spike likely to exist around the M87
supermassive black hole, combined with the high resolution observations of the
EHT Collaboration, can lead to strong limits on the primordial black hole mass
fraction in an astrophysically relevant mass range. The results depend on the
model assumed for the dark matter spike and suggest the interest of further
understanding of such spikes as well as further high resolution observations
on supermassive black holes.

##  1 Introduction

Ascertaining the nature of dark matter remains one of the outstanding problems
in cosmology. In recent times there have been active discussions of primordial
black holes (PBH’s) as a possible dark matter candidate or as a component
thereof. These discussions are motivated both by the standard Lambda Cold Dark
Matter (LCDM) model of cosmology, which can plausibly generates PBHs in
abundance, and by the absence of any need to postulate new physics in order to
establish their existence [1] .

There have been numerous studies that constrain the mass range of primordial
black holes as a significant contributor to dark matter. These include
gravitational microlensing [2] and ultrafaint dwarf galaxy heating [3].
However none of these limits are conclusive, either because of the effects of
PBH clustering or because of uncertainties in dwarf galaxy dynamical modelling
[4]. Indeed, it has equally been argued that the core/cusp transition in
dwarfs may be due to PBH heating of cold dark matter [5]. Here we consider a
new limit that enables us to set independent constraints on the fraction of
asteroid to subsolar or even solar mass PBHs that can currently be a
significant dark matter (DM) contributor.

##  2 Limits on resolution due to black holes

If there is a high density of black holes in some region of space the
gravitational fields will be ‘lumpy’ at short distances, inducing random
deflections of light rays. These deflections lead to a limitation on the
angular resolution possible in observations, and if a high resolution is
exhibited in some observation, there is then a limit on the presence of black
holes along the flight path. In ref [6] this effect was considered for rays
originating from the cosmic microwave background (CMB) and in ref [7] a
related, potentially more sensitive method, applicable even when there are no
small angle features to observe, was suggested in terms of the loss of
coherence between the arms of an interferometer,

However, for the application presented here, the first method, in terms of the
angular resolution, is the more favorable. The relevant formula for the limit
[6] on the possible angular resolution $\delta\alpha_{lim}$ for rays passing
through a column density of dark matter $\Sigma$ is

$(\delta\alpha_{lim})^{2}\approx{5}\times{10^{-2}}(M/M_{\odot})\,\Sigma_{ly}={5}\times{10^{-3}}(M/M_{\odot})\Sigma_{pc}\,,$
(1)

where the dark matter is composed of black holes of mass $M$. The angle
$\delta\alpha_{lim}$ is in $\mu\rm arcsec$ and $\Sigma$ on the right is in
terms of solar masses per $\rm pc^{2}$, as we will use in the following.

Eq 1 is based on the reasoning that if a photon has a high probability,
approaching one, of being scattered by a certain angle or more during its
flight, then an angular resolution of this amount is not possible. The
probability of such a scattering is found from the product of the column
density with the cross-section corresponding to the formula
$\delta\alpha=2r_{s}/b$ ($r_{s}$=Schwarzschild radius $=2GM,$ $b$ is impact
parameter of the photon , $c=1$ units), as in the bending of light by the sun.

A situation combining very high resolution observations with a suspected high
density of dark matter arises for the Super Massive Black Hole (SMBH) in the
galaxy M87. The EHT collaboration has studied thia object with a resolution of
10 $\mu\rm arcsec$ [8]. At the same time it is likely [9] that a SMBH is
surrounded by a “spike” of dark matter. In the present note, we would like to
consider the implications of Eq 1 in this situation.

##  3 Models around SMBHs: M87

To estimate $\Sigma$, a knowledge of the spike density profile is necessary.
This profile is generated by adiabatic growth of the SMBH at early epochs. Its
slope however depends on the growth history.

We consider two alternative density profiles. One is $\rho\propto r^{-7/3}$
for adiabatic growth of the SMBH by gas accretion or stellar tidal disruption.
The ambient cold dark matter responds adiabatically and develops a density
spike in the zone of influence of the SMBH. For the present considerations,
the dark matter specifically consists of primordial black holes, over what we
find to be an astrophysically relevant mass range.

A more conservative case appeals to dynamical relaxation of the accreted dark
matter. This would occur after a major merger, although adiabatic growth
should eventually resume. In this case, $\rho\propto r^{-3/2}$ for a black
hole merging scenario [10]. A 3/2 profile also results from primordial black
hole scattering off of a nuclear star cluster [11].

For the model, we also need the radius $r_{spike}$ where the density spike
begins, within the radius of gravitational influence of the central
supermassive black hole, defined by the competition between SMBH gravity and
the central stellsr velocity dispersion of M87. The spike radius can be
defined as [9] the radius where the potential of the SMBH falls to that of the
inner galaxy, as infered from the stellar velocity dispersion via virial
arguments. One then has $r_{spike}=GM_{SMBH}/\sigma_{\ast}^{2},$ where the
black hole mass in M87 is [8] $M_{SMBH}=6.5\pm 0.7\times 10^{9}\rm\,M_{\odot}$
and the central stellar velocity dispersion [12] is (with our estimated error
from reported long slit absorption spectroscopy) $\sigma_{\ast}=400\pm\rm
40\,km/s.$ We thus take $r_{spike}=200\rm\,pc$.

###  3.1 Normalization of the spike

In addition to the density profile, we also need to evaluate the absolute size
or normalization of the spike. This can be represented by the value of the
density at $r_{spike}$, which we call $\rho_{spike}$. This can then be
extrapolated according to the assumed power law to find $\rho_{horizon}$ as
needed in Eq 3.2 below.

We assume a certain mass for the spike and relate this to $\rho_{spike}$.
Integrating the power law, one finds

$M_{spike}\approx 4\pi\rho_{spike}r_{spike}^{3}\frac{1}{3-n}\,,$ (2)

where $n$ is the power in the profile. One notes that this, as opposed to the
integral for $\Sigma$ below, is essentially independent of $n$. This is
because when $\rho$ is not too singular, $n<3$, the geometric factor $r^{2}$
in the integral makes it depend essentially on the outer scale.

To proceed, we parameterize the spike mass in terms of a constant $\eta$ as

$M_{spike}=\eta\times{1}\times{10^{9}}M_{\odot}\,,$ (3)

so that $\eta=1$ would correspond to $M_{spike}/M_{SMBH}=0.16$ for M87.. The
observed value of $\rho_{spike}$ can be inferred from the central kinematics
of M87 [13] to approximately constrain $\eta\sim 0.1$.

We thus write

$\rho_{spike}=\frac{(3-n)}{4\pi}\frac{1}{r_{spike}^{3}}M_{spike}=\eta\times{3}\times{10^{1}}(1-n/3)\rm
M_{\odot}/pc^{3}\,.$ (4)

We record these values for $\rho_{spike}$ in the second column of the Table.
The third column gives the density scaled by the respective power laws to
$r_{horizon}=0.001\,\rm pc$ and the fourth column the resulting column density
according to Eq 3.2 below.

###  3.2 Surface density through the “spike”

The column or surface density $\Sigma$ is given by the integral of the
ordinary density $\rho$ along the flight path: $\Sigma=\int\rho(r)dr$.
Although the integral should in principle be taken over the entire flight
path, it will be dominated by the ‘spike’ and so we estimate it simply from
the beginning of the spike, $r_{spike}$ to the vicinity of the SMBH,
$r_{horizon}$. We take the inner spike radius $r_{horizon}=0.001\,\rm pc$ from
general relativistic studies of the spike for Schwarzschild [14] or Kerr [15]
black holes.

Thus

$\Sigma\approx\int_{r_{spike}}^{r_{horizon}}\rho(r)dr$ (5)

With $\rho$ represented by the power law $\rho\sim r^{-n}$ we introduce the
rescaled radius $x=r/r_{horizon}$. Normalizing to the value at the horizon, we
can write

$\rho=\rho_{horizon}\frac{1}{x^{n}}\,.$ (6)

The value of $\rho_{horizon}$ can be found by magnifying the ambient dark
matter density at the beginning of the spike by the factor
$(r_{spike}/r_{horizon})^{n}$. It is this large factor which can make the
spikes significant in the calculation of a column density . We now carry out
the integral Eq 5 in the limit $(r_{horizon}/r_{spike})<<1$.

$\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle
r_{horizon}\times\rho_{horizon}\int_{1}^{(r_{spike}/r_{horizon})}\frac{1}{x^{n}}dx$
$\displaystyle=$
$\displaystyle\bigl{(}\frac{1}{n-1}\bigr{)}\,r_{horizon}\times\rho_{horizon}(1-(r_{horizon}/{r_{spike})^{n-1}})$
$\displaystyle\approx$
$\displaystyle\bigl{(}\frac{1}{n-1}\bigr{)}\,r_{horizon}\times\rho_{horizon}\,.$

The resulting values for $\Sigma$ are shown in the last column of the Table.

Model | $\rho_{spike}$ | $\rho_{horizon}$ | $\Sigma$
---|---|---|---
$\rho\sim r^{-3/2}$ | $\eta\times 15$ | $\eta\times{1}\times{10^{9}}$ | $\eta\times{3}\times{10^{6}}$
|  |  |
$\rho\sim r^{-7/3}$ | $\eta\times 6.5$ | $\eta\times{2}\times{10^{13}}$ | $\eta\times{1}\times{10^{10}}$
|  |  |
Table 1: Parameters for two models of the dark matter spike at the SMBH of
M87. The beginning of the spike, $r_{spike}$, has been taken to be $200pc$
from the black hole and its inner radius $r_{horizon}$ near the SMBH’s
horizon, namely $0.001pc$. The parameter $\eta$ characterizes the mass in the
spike via Eq 3. The units of the dark matter density $\rho$ are $\rm
M_{\odot}/pc^{3}$ and the units of the column or surface density $\Sigma$ are
$\rm M_{\odot}/pc^{2}$.

##  4 Application of EHT resolution of 10 $\mu\rm arcsec$

With these estimates for $\Sigma$, we can apply Eq 1 using the resolution of
10 $\mu\rm arcsec$ stated by the EHT collaboration [8]. Inserting 10 $\mu\rm
arcsec$ on the lhs of Eq 1 and requiring that the rhs not exceed this, one
finds, with $\Sigma$ in units of $\rm M_{\odot}/pc^{2}$

$\Sigma\times(\rm M/M_{\odot})<{2}\times{10^{4}}$ (8)

The possibility is commonly entertained that the dark matter is composed only
partially of primordial black holes, characterized as a fraction $f_{bh}$ of
the total dark matter mass.

In the present case, we should then use a column density reduced by $f_{bh}$.
Since the dark matter mass enters linearly everywhere, one could effect this,
for example, by making the replacement $\eta\to\eta\times f_{bh}$ in the
formulae. Using the inferred value of $\Sigma$ for the two models, one has the
limits

$\displaystyle M\times f_{pbh}\leq\frac{1}{\eta}\times{7}\times{10^{-3}}\rm
M_{\odot}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ $\displaystyle\rho\sim
r^{-3/2}\,model$ (9) $\displaystyle M\times
f_{pbh}\leq\frac{1}{\eta}\times{2}\times{10^{-6}}\rm
M_{\odot}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ $\displaystyle\rho\sim
r^{-7/3}\,model$

These limits should apply regardless of the possible clustering of primordial
black holes. Clusters would not survive within the density spike around the
SMBH. Poisson fluctuation-induced clustering has been shown to greatly reduce
the number of predicted merger events [16], and allows the resurrection of a
significant solar mass component of PBHs that could account for the observed
LIGO/VIRGO black hole merger rates [17]. The astrophysical and primordial
black hole interpretationa are currently indistinguishable [18]. Our results
show the possibiity for strong limits on compact objects by a new independent
method. In particular the second variant in Eq 9 is of interest for the
asteroid/earth mass range. This can be compared with the results of the OGLE
[19] and Subaru [20] microlensing collaborations. The M31 microlensing data,
revised for finite lens size effects [21], sets a limit of around 1% on the
mass fraction of PBHs contributing to dark matter in the mass range
$10^{-9}-10^{-6}\rm M_{\odot}.$

##  5 Comparisons

A number of other methods have been used to limit the possibilities for
primordial black holes . In Fig 1 we show some representative limits, in the
$(M/M_{\odot},f_{pbh})$ plane, as compared to ours from Eq 9. We show the case
$\eta=0.1$. The dashed (right) line is for the $\rho\sim r^{-3/2}$ model and
the solid (left) line is for the $\rho\sim r^{-7/3}$ model. The excluded
region, according to Eq 9, lies to the right and above these lines.

Our arguments appear to exclude a considerable region of the right-hand part
of the $(M/M_{\odot},f_{pbh})$ plane, in agreement with, and with the
$\rho\sim r^{-7/3}$ model, extending previous exclusions. On the other hand,
in the limiting case where the dark matter consists entirely of primordial
black holes or $f_{pbh}=1$, the method does not seem to provide limits as
strong as those from microlensing. Improvement in the input information, such
as higher resolution observations, would tend to move the exclusion lines to
the left.

The other exclusion limits shown on the plot have been obtained from reference
[1] and from reference [22].

Figure 1: Limits on primordial black holes exhibited in the
$(M/M_{\odot},f_{pbh})$ plane, using Eq 9, for the $\rho\sim r^{-3/2}$ model (
dashed, right, line) and for the $\rho\sim r^{-7/3}$ model (solid, left, line)
The respective excluded regions are to the right of these lines. Other limits
are quoted from [1] and [22]. Figure graciously provided by Guillem Domenech.

##  6 Interpretation

These interesting limits illustrate the potential power of the method. They
arise essentially from the sharpness of the spike, with the associated large
values of $\rho_{horizon}$. The large difference between the two variants in
Eq 9 arises from the sensitivity to the steepness of the spike profile. Thus
an improvement in the definition of our results would come from a better
understanding of the spike profile, either by observational or theoretical
arguments. Of course, better knowledge of $M_{spike}$ or even better
observational resolution would also be helpful.

As stresed in [6] and [7] the interpretation of a possible positive signal
would be a much more subtle matter that the simple setting of limits as we do
here. That is, if a loss of, or a limit on, observational resolution is found,
all other reasons for this, natural or instrumental, have to be considered
before attributing the effect to primordial black holes . Further observations
of other SMBH’s with high resolution would be of interest in connection with
our arguments. Note that in Eq 1 the resolution enters quadratically, and
similarly on the rhs of Eq 8, so improved high resolution will have a strong
effect on the limits.

It is difficult to avoid the development of a dark matter spike around the
SMBH in M87. It may be modulated by SMBH merging, but this would have most
likely occurred within the first billion years after SMBH formation. Hence the
effective value of accumulated DM mass in the spike could be slightly reduced.
We argue that the adiabatic build-up of the central SMBH in massive elliptical
galaxies inevitably results in DM spike formation, with intriguing
consequences for the EHT observations of the supermassive black hole in M87.

In the event that PBHs form all or most of the DM, an interesting aspect of
our method is that any pre-existing clustering of PBHs, such as that invoked
to account for consistency with the early MACHO experiments and with the
LIGO/VIRGO event rates, would not be preserved in the spike. With improved
understanding of spikes and further observations involving them, our new kind
of limit promises new, independent constraints on PBH masses as a DM
contributor.

##  7 Acknowledgement

We thank Guillem Domenech for providing us with Figure 1.

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