EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

CERN-EP-2016-189 LHCb-PAPER-2016-029 6 September 2016

Search for structure in the ${{B}^{0}_{s}}{{\pi}^{\pm}}$ invariant mass
spectrum

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

The ${{B}^{0}_{s}}{{\pi}^{\pm}}$ invariant mass distribution is investigated
in order to search for possible exotic meson states. The analysis is based on
a data sample recorded with the LHCb detector corresponding to
$3\mbox{\,fb}^{-1}$ of $pp$ collision data at $\sqrt{s}=7$ and
$8\mathrm{\,Te\kern-1.00006ptV}$. No significant excess is found, and upper
limits are set on the production rate of the claimed $X(5568)$ state within
the LHCb acceptance. Upper limits are also set as a function of the mass and
width of a possible exotic meson decaying to the ${{B}^{0}_{s}}{{\pi}^{\pm}}$
final state. The same limits also apply to a possible exotic meson decaying
through the chain ${{B}^{*0}_{s}}{{\pi}^{\pm}}$,
${{B}^{*0}_{s}}\rightarrow{{B}^{0}_{s}}\gamma$ where the photon is excluded
from the reconstructed decays.

Published in Phys. Rev. Lett. 117 (2016) 152003

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

Interest in exotic hadrons has recently intensified, with a wealth of
experimental data becoming available [1, 2]. All the well-established exotic
states contain a heavy quark-antiquark ($c\bar{c}$ or $b\bar{b}$) pair
together with additional light particle content. However, the D0 collaboration
has reported evidence [3] of a narrow structure, referred to as the $X(5568)$,
in the ${{B}^{0}_{s}}{{\pi}^{\pm}}$ spectrum produced in $p\bar{p}$ collisions
at centre-of-mass energy $\sqrt{s}=1.96\mathrm{\,Te\kern-1.00006ptV}$. The
claimed $X(5568)$ state, if confirmed, would differ from any of the previous
observations, as it must have constituent quarks with four different flavours
($b$, $s$, $u$, $d$). As such, it would be unique among observed exotic
hadrons in having its mass dominated by a single constituent quark rather than
by a quark-antiquark pair. This could provide a crucial piece of information
to help understand how exotic hadrons are bound; specifically, whether they
are dominantly tightly bound (often referred to as “tetraquarks” and
“pentaquarks”) or loosely bound meson-meson or meson-baryon molecules.

In this Letter, results are presented from a search for an exotic meson,
denoted $X$, decaying to ${{B}^{0}_{s}}{{\pi}^{\pm}}$ in a data sample
corresponding to $3\mbox{\,fb}^{-1}$ of $pp$ collision data at $\sqrt{s}=7$
and $8\mathrm{\,Te\kern-1.00006ptV}$ recorded by LHCb. The search is performed
by scanning over the mass and width of the purported state, with dedicated
fits for parameters corresponding to those of the claimed $X(5568)$ state. The
${{B}^{0}_{s}}$ mesons are reconstructed in decays to
${{D}^{-}_{s}}{{\pi}^{+}}$ and ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ final states to obtain a ${B}^{0}_{s}$ yield approximately 20
times larger than that used by the D0 collaboration. The inclusion of charge-
conjugate processes is implied throughout the Letter. The analysis techniques
follow closely those developed for studies of the ${{{B}^{+}}}{{K}^{-}}$ [4],
${{{B}^{+}}}{{\pi}^{-}}$ and ${{B}^{0}}{{\pi}^{+}}$ [5] spectra. As in
previous analyses, the charged pion which is combined with the ${B}^{0}_{s}$
meson in order to form the ${{B}^{0}_{s}}{{\pi}^{\pm}}$ candidate is referred
to as the “companion pion”.

The LHCb detector [6, 7] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with 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 momentum, $p$, of
charged particles with a relative uncertainty that varies from $0.5\,\%$ at
low momentum to $1.0\,\%$ at $200\mathrm{\,Ge\kern-1.00006ptV}$ (units in
which $c=\hbar=1$ are used throughout). The minimum distance of a track to a
primary vertex (PV), the impact parameter, is measured with a resolution of
$(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in
$\mathrm{\,Ge\kern-1.00006ptV}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The online event selection is performed by a trigger, which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction.

Simulations of $pp$ collisions are generated using Pythia [8,
*Sjostrand:2007gs] with a specific LHCb configuration [10]. Decays of hadronic
particles are described by EvtGen [11], in which final-state radiation is
generated using Photos [12]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [13,
*Agostinelli:2002hh] as described in Ref. [15].

Candidate ${{B}^{0}_{s}}$ mesons are reconstructed through the decays
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ with
${{D}^{-}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{-}}$, and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}\mu^{-}}$ and $\phi\rightarrow{{K}^{+}}{{K}^{-}}$.
Particle identification, track quality and impact parameter requirements are
imposed on all final-state particles. Both ${B}^{0}_{s}$ and intermediate
particle (${{D}^{-}_{s}}$ and ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$) candidates are required to have good vertex quality and to have
invariant mass close to the known values [16]. Specific backgrounds due to
other $b$-hadron decays are removed with appropriate vetoes. A requirement is
imposed on the multiplicity of tracks originating from the PV associated with
the ${B}^{0}_{s}$ candidate; this requirement is about $90\,\%$ efficient on
${B}^{0}_{s}$ signal and significantly reduces background due to random
${{B}^{0}_{s}}{{\pi}^{\pm}}$ combinations. To further reduce background, the
$p_{\mathrm{T}}$ of the ${B}^{0}_{s}$ candidate,
$\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})$, is required to be greater than
$5\mathrm{\,Ge\kern-1.00006ptV}$. Results are also obtained with this
requirement increased to $10$ or $15\mathrm{\,Ge\kern-1.00006ptV}$, to be more
sensitive to scenarios in which the $X$ state is predominantly produced from
hard processes. The definition of the fiducial acceptance is completed with
the requirements
$\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})<50\mathrm{\,Ge\kern-1.00006ptV}$ and
$2.0<y<4.5$, where $y$ is the rapidity of the ${B}^{0}_{s}$ candidate.

The signals in the two ${B}^{0}_{s}$ decay modes are shown in Fig. 1. To
estimate the ${B}^{0}_{s}$ yields, the data are fitted with functions that
include a signal component, described by a double Gaussian function with a
shared mean, and a combinatorial background component, described by a
polynomial function. Backgrounds from
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ decays in the
${{D}^{-}_{s}}{{\pi}^{+}}$ sample and from ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}{{K}^{-}}$ decays in the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ sample, where a final-state hadron is misidentified, are modelled
using empirical shapes derived from simulated samples. An additional
component, modelled with a Gaussian function, is included to account for
possible ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays [17] in the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ sample. The results of
these fits are reported in Table 1. The signal-to-background ratio in the
${B}^{0}_{s}$ signal windows is about 10 for the ${{D}^{-}_{s}}{{\pi}^{+}}$
sample and above 50 for the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ sample.

Figure 1: Selected candidates for (left)
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ and (right)
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
decays, with
$\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>5\mathrm{\,Ge\kern-0.90005ptV}$, where
the ${B}^{0}_{s}$ signal window requirements of
$|m({{D}^{-}_{s}}{{\pi}^{+}})-5367\mathrm{\,Me\kern-0.90005ptV}|<30\mathrm{\,Me\kern-0.90005ptV}$
and $|m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)-5367\mathrm{\,Me\kern-0.90005ptV}|<13\mathrm{\,Me\kern-0.90005ptV}$
are indicated by dotted lines. Results of the fits described in the text are
superimposed with the total fit result shown as a red line, the signal
component as an unfilled area, the combinatorial background component as a
dark blue area and additional background contributions as a light green area.

The ${B}^{0}_{s}$ candidates are combined with each track originating from the
associated PV that gives a good quality ${{B}^{0}_{s}}{{\pi}^{\pm}}$ vertex
and that has $\mbox{$p_{\mathrm{T}}$}>500\mathrm{\,Me\kern-1.00006ptV}$. A
loose pion identification requirement is imposed in order to suppress possible
backgrounds involving misidentified particles. In case multiple candidates are
obtained in the same event, all are retained. Mass and vertex constraints are
imposed [18] in the calculation of the ${{B}^{0}_{s}}{{\pi}^{\pm}}$ invariant
mass.

In order to obtain quantitative results on the contributions from resonant
structures in the data, the ${{B}^{0}_{s}}{{\pi}^{\pm}}$ mass distributions
are fitted with a function containing components for signal and background.
The signal shape is an S-wave Breit–Wigner function multiplied by a function
that accounts for the variation of the efficiency with
${{B}^{0}_{s}}{{\pi}^{\pm}}$ mass. The efficiency function, determined from
simulation, plateaus at high ${{B}^{0}_{s}}{{\pi}^{\pm}}$ mass and falls near
the threshold to a value that depends on
$\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})$. The resolution is better than
$1\mathrm{\,Me\kern-1.00006ptV}$ and does not affect the results.The
background is modelled with a polynomial function. It is verified that this
function gives a good description of backgrounds composed of either a real or
a fake ${B}^{0}_{s}$ decay combined with a random pion, as determined from
simulation or from data in ${B}^{0}_{s}$ candidate mass sideband regions,
respectively.

For each choice of signal mass and width parameters, a binned maximum
likelihood fit to the ${{B}^{0}_{s}}{{\pi}^{\pm}}$ candidate mass spectrum is
used to determine the signal and background yields and the parameters of the
polynomial shape that describes the background. The two ${B}^{0}_{s}$ decay
modes are fitted simultaneously. The results of the fit where the mass and
width are fixed according to the central values obtained by the D0
collaboration, $m=5567.8\pm
2.9\mathrm{\,(stat)}\,^{+0.9}_{-1.9}\mathrm{\,(syst)}\mathrm{\,Me\kern-1.00006ptV}$
and $\Gamma=21.9\pm
6.4\mathrm{\,(stat)}\,^{+5.0}_{-2.5}\mathrm{\,(syst)}\mathrm{\,Me\kern-1.00006ptV}$
[3], are shown in Fig. 2 for both ${B}^{0}_{s}$ decay modes combined. The
$X(5568)$ yield is not significant for any minimum
$\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})$ requirement. In each case the change
in negative log-likelihood between fits including or not including the signal
component is less than 2 units for two additional free parameters
corresponding to the yields in the two ${B}^{0}_{s}$ decay modes. The results
of the fits are summarised in Table 1.

Figure 2: Results of the fit to the ${{B}^{0}_{s}}{{\pi}^{\pm}}$ mass
distribution for candidates (both ${B}^{0}_{s}$ modes combined) with minimum
$\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})$ of (top)
$5\mathrm{\,Ge\kern-0.90005ptV}$, (middle) $10\mathrm{\,Ge\kern-0.90005ptV}$
and (bottom) $15\mathrm{\,Ge\kern-0.90005ptV}$. The component for the claimed
$X(5568)$ state is included in the fit but is not significant. The
distributions of the normalised residuals, or “pulls”, displayed underneath
the main figures show good agreement between the fit functions and the data.

The yields $N$ obtained from the fits are used to measure the ratio of cross-
sections

$\displaystyle\rho_{X}^{\rm LHCb}$ $\displaystyle\equiv$
$\displaystyle\frac{\sigma(pp\rightarrow X+{\rm anything})\times{\cal
B}\left(X\rightarrow{{B}^{0}_{s}}{{\pi}^{\pm}}\right)}{\sigma(pp\rightarrow{{B}^{0}_{s}}+{\rm
anything})}\,,$ (1) $\displaystyle=$
$\displaystyle\frac{N(X)}{N({{B}^{0}_{s}})}\times\frac{1}{\epsilon^{\rm
rel}(X)}\,,$ (2)

where the cross-sections $\sigma$ are for promptly produced particles within
the LHCb acceptance. Since $\sigma(pp\rightarrow{{B}^{0}_{s}}+{\rm anything})$
in the LHCb acceptance has been previously measured [19], any result for
$\rho_{X}^{\rm LHCb}$ can be scaled to give a result for $\sigma(pp\rightarrow
X+{\rm anything})\times{\cal
B}\left(X\rightarrow{{B}^{0}_{s}}{{\pi}^{\pm}}\right)$ in the LHCb acceptance.
The relative efficiency $\epsilon^{\rm
rel}(X)=\frac{\epsilon(X)}{\epsilon({{B}^{0}_{s}})}$ accounts for the
reconstruction and selection efficiency of the companion pion as well as the
requirement that it is within the LHCb acceptance. These effects are
determined from simulation, weighted to reproduce the measured differential
${B}^{0}_{s}$ production spectrum [19], together with a data-driven evaluation
[20] of the efficiency of the particle identification requirement on the
companion pion. In the simulation the $X$ state is assumed to be spinless; it
has been verified that the systematic uncertainty associated with this choice
is negligible. The quantities used to evaluate $\rho_{X}^{\rm LHCb}$ are
summarised in Table 1.

Table 1: Yields, $N$, of ${B}^{0}_{s}$ and $X(5568)$ candidates obtained from the fits to the ${B}^{0}_{s}$ and ${{B}^{0}_{s}}{{\pi}^{\pm}}$ candidate mass distributions, with statistical uncertainties only. The values reported for $N({{B}^{0}_{s}})$ are those inside the ${B}^{0}_{s}$ signal window. The reported values for $X(5568)$ are obtained from fits with signal mass and width parameters fixed to those determined by the D0 collaboration. Relative efficiencies $\epsilon^{\rm rel}(X)$ of the ${B}^{0}_{s}$ and $X(5568)$ candidate selection criteria are also given. The reported uncertainties on the relative efficiencies are only statistical, due to the finite size of the simulated samples.  |  | ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ | ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ |  $-$ Sum
---|---|---|---|---
| $\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>\phantom{1}5\mathrm{\,Ge\kern-1.00006ptV}$ | 62.2$\,\pm\,$ | 0.3 | 43.6$\,\pm\,$ | 0.2 | 105.8$\,\pm\,$ | 0.4
$N({{B}^{0}_{s}})/10^{3}$ | $\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>10\mathrm{\,Ge\kern-1.00006ptV}$ | 28.4$\,\pm\,$ | 0.2 | 13.2$\,\pm\,$ | 0.1 | 41.6$\,\pm\,$ | 0.2
| $\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>15\mathrm{\,Ge\kern-1.00006ptV}$ | 8.8$\,\pm\,$ | 0.1 | 3.7$\,\pm\,$ | 0.1 | 12.5$\,\pm\,$ | 0.1
| $\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>\phantom{1}5\mathrm{\,Ge\kern-1.00006ptV}$ | 3$\,\pm\,$ | 64 |  $-$33$\,\pm\,$ | 43 |  $-$30$\,\pm\,$ | 77
$N(X)$ | $\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>10\mathrm{\,Ge\kern-1.00006ptV}$ | 75$\,\pm\,$ | 52 | 12$\,\pm\,$ | 33 | 87$\,\pm\,$ | 62
| $\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>15\mathrm{\,Ge\kern-1.00006ptV}$ | 14$\,\pm\,$ | 31 |  $-$10$\,\pm\,$ | 17 | 4$\,\pm\,$ | 35
| $\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>\phantom{1}5\mathrm{\,Ge\kern-1.00006ptV}$ | 0.127$\,\pm\,$ | 0.002 | 0.093$\,\pm\,$ | 0.001 |  $-$ —
$\epsilon^{\rm rel}(X)$ | $\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>10\mathrm{\,Ge\kern-1.00006ptV}$ | 0.213$\,\pm\,$ | 0.003 | 0.206$\,\pm\,$ | 0.002 |  $-$ —
| $\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>15\mathrm{\,Ge\kern-1.00006ptV}$ | 0.289$\,\pm\,$ | 0.005 | 0.290$\,\pm\,$ | 0.004 |  $-$ —

Systematic uncertainties are assigned due to possible biases in the evaluation
of $N(X)$, $N({{B}^{0}_{s}})$ and $\epsilon^{\rm rel}(X)$. The signal shape is
modified by varying the efficiency function, and separately by changing the
assumed angular momentum in the relativistic Breit–Wigner function from S-wave
to P-wave. In each case, the changes in $N(X)$ are assigned as the associated
systematic uncertainties. Uncertainties associated with the determination of
$N({{B}^{0}_{s}})$ arise due to the size of the ${B}^{0}_{s}$ sample and the
estimation of the background in the signal region. In addition to the limited
size of the simulation sample, uncertainties associated with $\epsilon^{\rm
rel}(X)$ arise due to the precision with which the companion pion
reconstruction and particle identification efficiencies are known [21, 20].
The uncertainties from different sources are combined in quadrature and give a
total that is much smaller than the statistical uncertainty. To obtain results
that can be compared to those for the claimed $X(5568)$ state reported by the
D0 collaboration, additional systematic uncertainties are assigned from the
changes in the results for $\rho_{X}^{\rm LHCb}$ when the mass and width
parameters are varied independently within $\pm 1\sigma$ ranges from their
central values. These are the dominant sources of systematic uncertainty.

To cross-check the results, candidates are selected with criteria similar to
those used in the observation of
${{B}_{c}^{+}}\rightarrow{{B}^{0}_{s}}{{\pi}^{+}}$ decays [22], with
consistent results. In addition, ${{B}^{0}}\rightarrow{{D}^{-}}{{\pi}^{+}}$
decays are used to create ${{B}^{0}}{{\pi}^{+}}$ combinations, and the results
on the excited $B$ states of Ref. [5] are reproduced.

The values of $\rho_{X}^{\rm LHCb}$ for the two ${B}^{0}_{s}$ decay modes are
consistent and are therefore combined in a weighted average. In the average,
systematic uncertainties are taken to be uncorrelated between the two
${B}^{0}_{s}$ decay modes. An exception is made when obtaining results
corresponding to the claimed $X(5568)$ state, where the uncertainty due to the
limited precision of the reported mass and width values [3] is treated as
correlated between the two modes. These results are

$\displaystyle\rho_{X}^{\rm
LHCb}(\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>\phantom{1}5\mathrm{\,Ge\kern-1.00006ptV})$
$\displaystyle=$ $\displaystyle-0.003\pm 0.006\pm 0.002\,,$
$\displaystyle\rho_{X}^{\rm
LHCb}(\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>10\mathrm{\,Ge\kern-1.00006ptV})$
$\displaystyle=$ $\displaystyle\phantom{-}0.010\pm 0.007\pm 0.005\,,$
$\displaystyle\rho_{X}^{\rm
LHCb}(\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>15\mathrm{\,Ge\kern-1.00006ptV})$
$\displaystyle=$ $\displaystyle\phantom{-}0.000\pm 0.010\pm 0.006\,,$

where the first uncertainty is statistical and the second is systematic. Since
the signal is not significant, upper limits on $\rho_{X}^{\rm LHCb}$ are
obtained by integration of the likelihood in the positive region to find the
value that contains the fraction of the integral corresponding to the required
confidence level (CL). The upper limits at 90 (95) % CL are found to be

$\displaystyle\rho_{X}^{\rm
LHCb}(\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>\phantom{1}5\mathrm{\,Ge\kern-1.00006ptV})$
$\displaystyle<$ $\displaystyle 0.011\ (0.012)\,,$ $\displaystyle\rho_{X}^{\rm
LHCb}(\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>10\mathrm{\,Ge\kern-1.00006ptV})$
$\displaystyle<$ $\displaystyle 0.021\ (0.024)\,,$ $\displaystyle\rho_{X}^{\rm
LHCb}(\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})>15\mathrm{\,Ge\kern-1.00006ptV})$
$\displaystyle<$ $\displaystyle 0.018\ (0.020)\,.$

No significant signal for a ${{B}^{0}_{s}}{{\pi}^{\pm}}$ resonance is seen at
any value of mass and width in the range considered. To obtain limits on
$\rho_{X}^{\rm LHCb}$ for different values of these parameters, fits are
performed for widths ($\Gamma$) of $10$ to $50\mathrm{\,Me\kern-1.00006ptV}$
in $10\mathrm{\,Me\kern-1.00006ptV}$ steps. For each width, the mass is
scanned in steps of $\Gamma/2$, starting one unit of width above the kinematic
threshold and ending approximately one and a half units of width below
$6000\mathrm{\,Me\kern-1.00006ptV}$. The upper edge of the range is chosen
because an exotic state with higher mass would be expected to give a clearer
signature in the ${{B}^{0}}{{K}^{\pm}}$ final state [23]. The results are
obtained in the same way as described above, and converted into upper limits
that are shown in Fig. 3. The upper limits are weaker when a broader width is
assumed, due to the larger amount of background under the putative peak. The
limits also become weaker when there is an excess of events in the signal
region, although all such excesses are consistent with being statistical
fluctuations. The method used to set the upper limits smooths out any negative
fluctuations.

Figure 3: Upper limits (ULs) at 90 % confidence level (CL) as functions of the
mass and width of a purported exotic state $X$ decaying to
${{B}^{0}_{s}}{{\pi}^{\pm}}$ with minimum
$\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})$ of (top)
$5\mathrm{\,Ge\kern-0.90005ptV}$, (middle) $10\mathrm{\,Ge\kern-0.90005ptV}$
and (bottom) $15\mathrm{\,Ge\kern-0.90005ptV}$. The same limits also apply to
a possible exotic meson decaying through the chain
${{B}^{*0}_{s}}{{\pi}^{\pm}}$, ${{B}^{*0}_{s}}\rightarrow{{B}^{0}_{s}}\gamma$
where the photon is excluded from the reconstructed decays. In the latter case
the nominal mass difference
$m({{B}^{*0}_{s}})-m({{B}^{0}_{s}})=48.6^{+1.8}_{-1.6}\mathrm{\,Me\kern-0.90005ptV}$
[16] has to be added to the values on the x-axis to get the mass of the exotic
meson under investigation.

In summary, a search for the claimed $X(5568)$ state has been carried out
using a data sample corresponding to $3~{}\mbox{\,fb}^{-1}$ of $pp$ collision
data at $\sqrt{s}=7$ and $8~{}\mathrm{\,Te\kern-1.00006ptV}$ recorded by LHCb.
No significant excess is found and thus the existence of the $X(5568)$ state
is not confirmed. Upper limits are set on the relative production rate of the
claimed state in the LHCb acceptance. Limits are also set as a function of the
mass and width of a possible exotic meson decaying to the
${{B}^{0}_{s}}{{\pi}^{\pm}}$ final state. The same limits also apply to a
possible exotic meson decaying through the chain
${{B}^{*0}_{s}}{{\pi}^{\pm}}$, ${{B}^{*0}_{s}}\rightarrow{{B}^{0}_{s}}\gamma$
where the photon is excluded from the reconstructed decays.

## 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 and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex
LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany), EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-
Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex
LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal
Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust
(United Kingdom).

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LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar6, J. Albrecht10,
F. Alessio40, M. Alexander53, S. Ali43, G. Alkhazov31, P. Alvarez Cartelle55,
A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L. An41, L. Anderlini18,
G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B. Appleby56, F.
Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M. Artuso61, E.
Aslanides6, G. Auriemma26, M. Baalouch5, I. Babuschkin56, S. Bachmann12, J.J.
Back50, A. Badalov38, C. Baesso62, S. Baker55, W. Baldini17, R.J. Barlow56, C.
Barschel40, S. Barsuk7, W. Barter40, M. Baszczyk27, V. Batozskaya29, B.
Batsukh61, V. Battista41, A. Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24,
I. Bediaga1, L.J. Bel43, V. Bellee41, N. Belloli21,i, K. Belous37, I.
Belyaev32, E. Ben-Haim8, G. Bencivenni19, S. Benson43, J. Benton48, A.
Berezhnoy33, R. Bernet42, A. Bertolin23, F. Betti15, M.-O. Bettler40, M. van
Beuzekom43, Ia. Bezshyiko42, S. Bifani47, P. Billoir8, T. Bird56, A.
Birnkraut10, A. Bitadze56, A. Bizzeti18,u, T. Blake50, F. Blanc41, J. Blouw11,
S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36,w, N. Bondar31,40, W.
Bonivento16, A. Borgheresi21,i, S. Borghi56, M. Borisyak35, M. Borsato39, F.
Bossu7, M. Boubdir9, T.J.V. Bowcock54, E. Bowen42, C. Bozzi17,40, S. Braun12,
M. Britsch12, T. Britton61, J. Brodzicka56, E. Buchanan48, C. Burr56, A.
Bursche2, J. Buytaert40, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,i, M.
Calvo Gomez38,m, A. Camboni38, P. Campana19, D. Campora Perez40, D.H. Campora
Perez40, L. Capriotti56, A. Carbone15,e, G. Carboni25,j, R. Cardinale20,h, A.
Cardini16, P. Carniti21,i, L. Carson52, K. Carvalho Akiba2, G. Casse54, L.
Cassina21,i, L. Castillo Garcia41, M. Cattaneo40, Ch. Cauet10, G. Cavallero20,
R. Cenci24,t, M. Charles8, Ph. Charpentier40, G. Chatzikonstantinidis47, M.
Chefdeville4, S. Chen56, S.-F. Cheung57, V. Chobanova39, M. Chrzaszcz42,27, X.
Cid Vidal39, G. Ciezarek43, P.E.L. Clarke52, M. Clemencic40, H.V. Cliff49, J.
Closier40, V. Coco59, J. Cogan6, E. Cogneras5, V. Cogoni16,40,f, L.
Cojocariu30, P. Collins40, A. Comerma-Montells12, A. Contu40, A. Cook48, S.
Coquereau38, G. Corti40, M. Corvo17,g, C.M. Costa Sobral50, B. Couturier40,
G.A. Cowan52, D.C. Craik52, A. Crocombe50, M. Cruz Torres62, S. Cunliffe55, R.
Currie55, C. D’Ambrosio40, F. Da Cunha Marinho2, E. Dall’Occo43, J. Dalseno48,
P.N.Y. David43, A. Davis59, O. De Aguiar Francisco2, K. De Bruyn6, S. De
Capua56, M. De Cian12, J.M. De Miranda1, L. De Paula2, M. De Serio14,d, P. De
Simone19, C.T. Dean53, D. Decamp4, M. Deckenhoff10, L. Del Buono8, M.
Demmer10, D. Derkach35, O. Deschamps5, F. Dettori40, B. Dey22, A. Di Canto40,
H. Dijkstra40, F. Dordei40, M. Dorigo41, A. Dosil Suárez39, A. Dovbnya45, K.
Dreimanis54, L. Dufour43, G. Dujany56, K. Dungs40, P. Durante40, R.
Dzhelyadin37, A. Dziurda40, A. Dzyuba31, N. Déléage4, S. Easo51, M. Ebert52,
U. Egede55, V. Egorychev32, S. Eidelman36,w, S. Eisenhardt52, U.
Eitschberger10, R. Ekelhof10, L. Eklund53, Ch. Elsasser42, S. Ely61, S.
Esen12, H.M. Evans49, T. Evans57, A. Falabella15, N. Farley47, S. Farry54, R.
Fay54, D. Fazzini21,i, D. Ferguson52, V. Fernandez Albor39, A. Fernandez
Prieto39, F. Ferrari15,40, F. Ferreira Rodrigues1, M. Ferro-Luzzi40, S.
Filippov34, R.A. Fini14, M. Fiore17,g, M. Fiorini17,g, M. Firlej28, C.
Fitzpatrick41, T. Fiutowski28, F. Fleuret7,b, K. Fohl40, M. Fontana16,40, F.
Fontanelli20,h, D.C. Forshaw61, R. Forty40, V. Franco Lima54, M. Frank40, C.
Frei40, J. Fu22,q, E. Furfaro25,j, C. Färber40, A. Gallas Torreira39, D.
Galli15,e, S. Gallorini23, S. Gambetta52, M. Gandelman2, P. Gandini57, Y.
Gao3, L.M. Garcia Martin68, J. García Pardiñas39, J. Garra Tico49, L.
Garrido38, P.J. Garsed49, D. Gascon38, C. Gaspar40, L. Gavardi10, G. Gazzoni5,
D. Gerick12, E. Gersabeck12, M. Gersabeck56, T. Gershon50, Ph. Ghez4, S.
Gianì41, V. Gibson49, O.G. Girard41, L. Giubega30, K. Gizdov52, V.V.
Gligorov8, D. Golubkov32, A. Golutvin55,40, A. Gomes1,a, I.V. Gorelov33, C.
Gotti21,i, M. Grabalosa Gándara5, R. Graciani Diaz38, L.A. Granado Cardoso40,
E. Graugés38, E. Graverini42, G. Graziani18, A. Grecu30, P. Griffith47, L.
Grillo21,40,i, B.R. Gruberg Cazon57, O. Grünberg66, E. Gushchin34, Yu. Guz37,
T. Gys40, C. Göbel62, T. Hadavizadeh57, C. Hadjivasiliou5, G. Haefeli41, C.
Haen40, S.C. Haines49, S. Hall55, B. Hamilton60, X. Han12, S. Hansmann-
Menzemer12, N. Harnew57, S.T. Harnew48, J. Harrison56, M. Hatch40, J. He63, T.
Head41, A. Heister9, K. Hennessy54, P. Henrard5, L. Henry8, J.A. Hernando
Morata39, E. van Herwijnen40, M. Heß66, A. Hicheur2, D. Hill57, C. Hombach56,
H. Hopchev41, W. Hulsbergen43, T. Humair55, M. Hushchyn35, N. Hussain57, D.
Hutchcroft54, M. Idzik28, P. Ilten58, R. Jacobsson40, A. Jaeger12, J.
Jalocha57, E. Jans43, A. Jawahery60, F. Jiang3, M. John57, D. Johnson40, C.R.
Jones49, C. Joram40, B. Jost40, N. Jurik61, S. Kandybei45, W. Kanso6, M.
Karacson40, J.M. Kariuki48, S. Karodia53, M. Kecke12, M. Kelsey61, I.R.
Kenyon47, M. Kenzie49, T. Ketel44, E. Khairullin35, B. Khanji21,40,i, C.
Khurewathanakul41, T. Kirn9, S. Klaver56, K. Klimaszewski29, S. Koliiev46, M.
Kolpin12, I. Komarov41, R.F. Koopman44, P. Koppenburg43, A. Kozachuk33, M.
Kozeiha5, L. Kravchuk34, K. Kreplin12, M. Kreps50, P. Krokovny36,w, F.
Kruse10, W. Krzemien29, W. Kucewicz27,l, M. Kucharczyk27, V. Kudryavtsev36,w,
A.K. Kuonen41, K. Kurek29, T. Kvaratskheliya32,40, D. Lacarrere40, G.
Lafferty56, A. Lai16, D. Lambert52, G. Lanfranchi19, C. Langenbruch9, T.
Latham50, C. Lazzeroni47, R. Le Gac6, J. van Leerdam43, J.-P. Lees4, A.
Leflat33,40, J. Lefrançois7, R. Lefèvre5, F. Lemaitre40, E. Lemos Cid39, O.
Leroy6, T. Lesiak27, B. Leverington12, Y. Li7, T. Likhomanenko35,67, R.
Lindner40, C. Linn40, F. Lionetto42, B. Liu16, X. Liu3, D. Loh50, I.
Longstaff53, J.H. Lopes2, D. Lucchesi23,o, M. Lucio Martinez39, H. Luo52, A.
Lupato23, E. Luppi17,g, O. Lupton57, A. Lusiani24, X. Lyu63, F. Machefert7, F.
Maciuc30, O. Maev31, K. Maguire56, S. Malde57, A. Malinin67, T. Maltsev36, G.
Manca7, G. Mancinelli6, P. Manning61, J. Maratas5,v, J.F. Marchand4, U.
Marconi15, C. Marin Benito38, P. Marino24,t, J. Marks12, G. Martellotti26, M.
Martin6, M. Martinelli41, D. Martinez Santos39, F. Martinez Vidal68, D.
Martins Tostes2, L.M. Massacrier7, A. Massafferri1, R. Matev40, A. Mathad50,
Z. Mathe40, C. Matteuzzi21, A. Mauri42, B. Maurin41, A. Mazurov47, M.
McCann55, J. McCarthy47, A. McNab56, R. McNulty13, B. Meadows59, F. Meier10,
M. Meissner12, D. Melnychuk29, M. Merk43, A. Merli22,q, E. Michielin23, D.A.
Milanes65, M.-N. Minard4, D.S. Mitzel12, A. Mogini8, J. Molina Rodriguez62,
I.A. Monroy65, S. Monteil5, M. Morandin23, P. Morawski28, A. Mordà6, M.J.
Morello24,t, J. Moron28, A.B. Morris52, R. Mountain61, F. Muheim52, M.
Mulder43, M. Mussini15, D. Müller56, J. Müller10, K. Müller42, V. Müller10, P.
Naik48, T. Nakada41, R. Nandakumar51, A. Nandi57, I. Nasteva2, M. Needham52,
N. Neri22, S. Neubert12, N. Neufeld40, M. Neuner12, A.D. Nguyen41, C. Nguyen-
Mau41,n, S. Nieswand9, R. Niet10, N. Nikitin33, T. Nikodem12, A. Novoselov37,
D.P. O’Hanlon50, A. Oblakowska-Mucha28, V. Obraztsov37, S. Ogilvy19, R.
Oldeman49, C.J.G. Onderwater69, J.M. Otalora Goicochea2, A. Otto40, P. Owen42,
A. Oyanguren68, P.R. Pais41, A. Palano14,d, F. Palombo22,q, M. Palutan19, J.
Panman40, A. Papanestis51, M. Pappagallo14,d, L.L. Pappalardo17,g, W.
Parker60, C. Parkes56, G. Passaleva18, A. Pastore14,d, G.D. Patel54, M.
Patel55, C. Patrignani15,e, A. Pearce56,51, A. Pellegrino43, G. Penso26, M.
Pepe Altarelli40, S. Perazzini40, P. Perret5, L. Pescatore47, K. Petridis48,
A. Petrolini20,h, A. Petrov67, M. Petruzzo22,q, E. Picatoste Olloqui38, B.
Pietrzyk4, M. Pikies27, D. Pinci26, A. Pistone20, A. Piucci12, S. Playfer52,
M. Plo Casasus39, T. Poikela40, F. Polci8, A. Poluektov50,36, I. Polyakov61,
E. Polycarpo2, G.J. Pomery48, A. Popov37, D. Popov11,40, B. Popovici30, S.
Poslavskii37, C. Potterat2, E. Price48, J.D. Price54, J. Prisciandaro39, A.
Pritchard54, C. Prouve48, V. Pugatch46, A. Puig Navarro41, G. Punzi24,p, W.
Qian57, R. Quagliani7,48, B. Rachwal27, J.H. Rademacker48, M. Rama24, M. Ramos
Pernas39, M.S. Rangel2, I. Raniuk45, G. Raven44, F. Redi55, S. Reichert10,
A.C. dos Reis1, C. Remon Alepuz68, V. Renaudin7, S. Ricciardi51, S.
Richards48, M. Rihl40, K. Rinnert54, V. Rives Molina38, P. Robbe7,40, A.B.
Rodrigues1, E. Rodrigues59, J.A. Rodriguez Lopez65, P. Rodriguez Perez56,†, A.
Rogozhnikov35, S. Roiser40, V. Romanovskiy37, A. Romero Vidal39, J.W.
Ronayne13, M. Rotondo19, M.S. Rudolph61, T. Ruf40, P. Ruiz Valls68, J.J.
Saborido Silva39, E. Sadykhov32, N. Sagidova31, B. Saitta16,f, V. Salustino
Guimaraes2, C. Sanchez Mayordomo68, B. Sanmartin Sedes39, R. Santacesaria26,
C. Santamarina Rios39, M. Santimaria19, E. Santovetti25,j, A. Sarti19,k, C.
Satriano26,s, A. Satta25, D.M. Saunders48, D. Savrina32,33, S. Schael9, M.
Schellenberg10, M. Schiller40, H. Schindler40, M. Schlupp10, M. Schmelling11,
T. Schmelzer10, B. Schmidt40, O. Schneider41, A. Schopper40, K. Schubert10, M.
Schubiger41, M.-H. Schune7, R. Schwemmer40, B. Sciascia19, A. Sciubba26,k, A.
Semennikov32, A. Sergi47, N. Serra42, J. Serrano6, L. Sestini23, P. Seyfert21,
M. Shapkin37, I. Shapoval45, Y. Shcheglov31, T. Shears54, L. Shekhtman36,w, V.
Shevchenko67, A. Shires10, B.G. Siddi17,40, R. Silva Coutinho42, L. Silva de
Oliveira2, G. Simi23,o, S. Simone14,d, M. Sirendi49, N. Skidmore48, T.
Skwarnicki61, E. Smith55, I.T. Smith52, J. Smith49, M. Smith55, H. Snoek43,
M.D. Sokoloff59, F.J.P. Soler53, B. Souza De Paula2, B. Spaan10, P.
Spradlin53, S. Sridharan40, F. Stagni40, M. Stahl12, S. Stahl40, P. Stefko41,
S. Stefkova55, O. Steinkamp42, S. Stemmle12, O. Stenyakin37, S. Stevenson57,
S. Stoica30, S. Stone61, B. Storaci42, S. Stracka24,p, M. Straticiuc30, U.
Straumann42, L. Sun59, W. Sutcliffe55, K. Swientek28, V. Syropoulos44, M.
Szczekowski29, T. Szumlak28, S. T’Jampens4, A. Tayduganov6, T. Tekampe10, M.
Teklishyn7, G. Tellarini17,g, F. Teubert40, E. Thomas40, J. van Tilburg43,
M.J. Tilley55, V. Tisserand4, M. Tobin41, S. Tolk49, L. Tomassetti17,g, D.
Tonelli40, S. Topp-Joergensen57, F. Toriello61, E. Tournefier4, S. Tourneur41,
K. Trabelsi41, M. Traill53, M.T. Tran41, M. Tresch42, A. Trisovic40, A.
Tsaregorodtsev6, P. Tsopelas43, A. Tully49, N. Tuning43, A. Ukleja29, A.
Ustyuzhanin35,67, U. Uwer12, C. Vacca16,f, V. Vagnoni15,40, A. Valassi40, S.
Valat40, G. Valenti15, A. Vallier7, R. Vazquez Gomez19, P. Vazquez Regueiro39,
S. Vecchi17, M. van Veghel43, J.J. Velthuis48, M. Veltri18,r, G. Veneziano41,
A. Venkateswaran61, M. Vernet5, M. Vesterinen12, B. Viaud7, D. Vieira1, M.
Vieites Diaz39, X. Vilasis-Cardona38,m, V. Volkov33, A. Vollhardt42, B.
Voneki40, A. Vorobyev31, V. Vorobyev36,w, C. Voß66, J.A. de Vries43, C.
Vázquez Sierra39, R. Waldi66, C. Wallace50, R. Wallace13, J. Walsh24, J.
Wang61, D.R. Ward49, H.M. Wark54, N.K. Watson47, D. Websdale55, A. Weiden42,
M. Whitehead40, J. Wicht50, G. Wilkinson57,40, M. Wilkinson61, M. Williams40,
M.P. Williams47, M. Williams58, T. Williams47, F.F. Wilson51, J. Wimberley60,
J. Wishahi10, W. Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton49, K.
Wraight53, S. Wright49, K. Wyllie40, Y. Xie64, Z. Xing61, Z. Xu41, Z. Yang3,
H. Yin64, J. Yu64, X. Yuan36,w, O. Yushchenko37, K.A. Zarebski47, M.
Zavertyaev11,c, L. Zhang3, Y. Zhang7, A. Zhelezov12, Y. Zheng63, A. Zhokhov32,
X. Zhu3, V. Zhukov9, S. Zucchelli15.

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 Savoie Mont-Blanc, 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
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Universidad de Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
65Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
66Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
67National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
68Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 38
69Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Viet Nam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vIligan Institute of Technology (IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
†Deceased


that, depending on the fixed value of $w$, one can have no roots, one root, or
two roots. The no root situation means that the space inside the heat
reservoir is hot de Sitter space. The one root situation means that there is
one possible black hole solution $y_{1}=y_{2}$ inside the reservoir. The two
roots situation means that there are two possible black hole solutions inside
the reservoir, one small $y_{1}$ and unstable, the other large $y_{2}$ and
stable. The case $x^{2}=1$ is important. When $x=1$ there is the sole solution
$y_{1}=y_{2}=1$ for $w=2$, but for any other $w$, $w>2$, there are two
solutions, one is $y_{1}$, corresponding to the small black hole, the other is
$y_{2}=1$, corresponding to the larger black holes, which for $x^{2}=1$ have
transubstantiated into the Nariai cosmological universes with all the allowed
values of $w$, $w\geq 2$.

###  VI.3 Further analysis: Cosmological horizons

In the Schwarzschild-de Sitter space there is also the radius of the
cosmological horizon $r_{\rm c}$. However, since the radius of the heat
reservoir $R$ is inside $r_{\rm c}$, this latter has no role in the
thermodynamics. The cosmological horizon radius, is just a parameter, which
has only a coadjuvant role in the whole setting. It is found once $r_{+}$ has
been found from the thermodynamics.

The radius of the cosmological horizon $r_{\rm c}$ is the largest root of the
equation $V(r)=0$, see Eq. (2) for the expression of $V(r)$. As we have seen,
it is related to the the black hole horizon radius according to $r_{\rm
c}^{2}+r_{\rm c}r_{+}+r_{+}^{2}-\frac{3}{\Lambda}=0$, whence $r_{\rm
c}=-\frac{r_{+}}{2}+\frac{r_{+}}{2}\sqrt{\frac{12-3\Lambda r_{+}^{2}}{\Lambda
r_{+}^{2}}}$, see Eq. (5), or $r_{\rm
c}=-\frac{r_{+}}{2}+\sqrt{\frac{3}{\Lambda}-\frac{3}{4}r_{+}^{2}}$. Let us
define

$u\equiv\frac{r_{\rm c}}{R}\,,$ (113)

as the cosmological radius in units of $R$. Then, from Eq. (5) we have that
$u$ is given in terms of $x$ and $y$ of Eqs. (82) and (83), respectively, by

$u=-\frac{y}{2}+\sqrt{\frac{3}{x^{2}}-\frac{3}{4}y^{2}}\,.$ (114)

Calculating $\frac{du}{dy}$ one finds

$\frac{du}{dy}\leq 0,$ (115)

i.e., $\frac{dr_{\rm c}}{dr_{+}}\leq 0$. This means that as the horizon radius
$r_{+}$ increases the cosmological horizon decreases, in general. We restrict
here to calculate some of the properties of the cosmological radius $u$.

We want to calculate the properties of $u$ at the neighborhood of the point
$x=1$, $y=1$, $u=1$, and find $\frac{du}{dx}$. Since this pertains to the
large black hole, we use, as usual, the subscript 2 to refer to that solution,
see Appendix C for details. One finds that

$\left(\frac{du_{2}}{dx}\right)_{x=1_{-}}=\left(\frac{dy_{2}}{dx}\right)_{x=1_{+}}$
(116)

and

$\left(\frac{du_{2}}{dx}\right)_{x=1_{+}}=\left(\frac{dy_{2}}{dx}\right)_{x=1_{-}}$
(117)

where $1_{-}$ and $1_{+}$ mean that one is taking the limit $x\to 1$ from
below, i.e., $x<1$, and from above, i.e., $x>1$, respectively. Using Eqs.
(96), (108), (116), and (117) we can deduce some further specific properties
of the cosmological horizon. They are that
$\left(\frac{du_{2}}{dx}\right)_{x=1_{-}}\to-\infty$ for $w\to 2$, and
$\left(\frac{du_{2}}{dx}\right)_{x=1_{+}}\to-2$ for $w\to\infty$, and that
$\left(\frac{du_{2}}{dx}\right)_{x=1_{-}}\to\infty$ for $w\to 2$, and
$\left(\frac{du_{2}}{dx}\right)_{x=1_{+}}\to 0$ for $w\to\infty$.

This interchange of equations as displayed in Eqs. (116) and (117), is mostly
clearly seen in the case $w=\infty$, i.e., $RT\to\infty$. Indeed, the case
$w\to\infty$ can be solved exactly as we have seen in Eq. (114). Thus, for
$0\leq x^{2}\leq 1$, one has $y_{2}=1$ and
$u_{2}=-\frac{1}{2}+\sqrt{\frac{3}{x^{2}}-\frac{3}{4}}$, whereas for $1\leq
x^{2}\leq 3$, $y_{2}=-\frac{1}{2}+\sqrt{\frac{3}{x^{2}}-\frac{3}{4}}$ and
$u_{2}=1$, see Fig. 5.

Figure 5:  Plots of $\frac{r_{+2}}{R}$ and $\frac{r_{\rm c2}}{R}$ as a
function of $\sqrt{\Lambda R^{2}}$ for $RT=\infty$, i.e., essentially infinite
reservoir temperature. For plotting purposes it is defined
$x\equiv\sqrt{\Lambda R^{2}}$, $y_{2}\equiv\frac{r_{+2}}{R}$,
$u_{2}\equiv\frac{r_{\rm c2}}{R}$, and $w\equiv 4\pi RT$. So this is the case
$w=\infty$. Note that $y_{2}$ and $u_{2}$ exchange character at the
bifurcation point $x=1$. See text for details.

In brief, the black hole horizon equation for $\Lambda R^{2}<1$ turns into the
cosmological horizon equation for $\Lambda R^{2}>1$, and the cosmological
horizon equation for $\Lambda R^{2}<1$ turns into the black hole horizon
equation for $\Lambda R^{2}>1$.

##  VII Conclusions

We have taken on the problem of understanding the Schwarzschild-de Sitter
black hole thermodynamically. For that we have put the black hole in a cavity
of radius $R$ surrounded by a heat reservoir and kept at temperature $T$. This
structure allowed the thermodynamic problem to be solved within the canonical
formalism and the Euclidean path integral to quantum gravity. Moreover, it led
naturally into two other spaces, namely, hot de Sitter and Nariai cosmological
spaces, besides the initial Schwarzschild-de Sitter black hole.

There are new results. One is that the range of the relevant parameter
$\Lambda R^{2}$ is extendable up to 3, i.e., $0\leq\Lambda R^{2}\leq 3$. It
was also found that to properly treat the problem one has to divide this range
into three ranges, $0\leq\Lambda R^{2}<1$, $\Lambda R^{2}=1$, $1<\Lambda
R^{2}\leq 3$.

On the lower side of $\Lambda R^{2}$, $0\leq\Lambda R^{2}<1$, York’s solution
for pure Schwarzschild is automatically incorporated when $\Lambda R^{2}=0$,
appearing first for $RT=\frac{\sqrt{27}}{8\pi}$, with a coincident black hole
horizon radius $r_{+1}=r_{+2}=\frac{2}{3}R$. For higher $\Lambda R^{2}$ the
coincident black hole horizon radius gets increased values for some lower
$RT$. A heuristic understanding of this behavior has been given. Changing the
values and $\Lambda R^{2}$ and $RT$ one obtains either two thermodynamics
solutions $r_{+1}$ small and $r_{+2}$ large, the first thermodynamically
unstable and the second stable, or no solutions in which case one is in the
presence of hot de Sitter.

At the intermediate value of $\Lambda R^{2}$, $\Lambda R^{2}=1$, the small
$r_{+1}$ unstable black hole exists. More interestingly, the large $r_{+2}$
black hole is now the solution $r_{+2}=R$ and opens out into a spectrum of a
beautiful set of Nariai universes that can have all temperatures in the range
$\frac{1}{2\pi}\leq RT\leq\infty$. The ensemble data $R$ and $T$ now determine
the location of the boundary $Z$ in the Nariai universe rather than the black
hole radius solution. The Nariai universe is thermodynamically neutrally
stable.

On the higher side of $\Lambda R^{2}$, $1<\Lambda R^{2}\leq 3$, unexpected
black hole solutions also arise. The small $r_{+1}$ unstable black hole exists
without changing character. The large $r_{+2}$ stable solutions interchange
the role of black hole $r_{+2}$ and cosmological $r_{\rm c2}$ horizons, and
the maximum value $r_{+2}$ can have is attained for infinite temperature,
i.e., $RT=\infty$, and is less than one changing as $\Lambda R^{2}$ changes up
to 3. The case at the end of the range, $\Lambda R^{2}=3$, only exists for
infinite temperature, and represents a reservoir filled with de Sitter space
inside, except at the center, where there is a black hole with zero horizon
radius $r_{+}=0$, i.e., a naked singularity.

Another important result is that increasing the temperature from zero, i.e.,
increasing $RT$ from zero, one finds that the first solution that appears for
the whole range $0\leq\Lambda R^{2}\leq 3$ has temperature $RT=\frac{1}{2\pi}$
and is a solution with $\Lambda R^{2}=1$, and radius $r_{+1}=r_{+2}=R$, i.e.,
it is the coincident Nariai solution. As one increases $RT$ solutions with
lower and higher $\Lambda R^{2}$ peal out.

Yet another interesting finding is related to phase transitions. In the
$\Lambda R^{2}<1$ case, in particular for $\Lambda R^{2}\ll 1$, the possible
thermodynamic phases that appear as one increases the temperature from zero
are hot de Sitter only, hot de Sitter favored in relation to the
Schwarzschild-de Sitter black hole, hot de Sitter coexisting equally with the
Schwarzschild-de Sitter black hole, the Schwarzschild-de Sitter black hole
favored in relation to hot de Sitter, and the Schwarzschild-de Sitter black
hole alone. This latter case comes out when one considers a high enough
temperature so that there is sufficient energy in the cavity to surpass the
Buchdahl bound and presumably the system collapses. There are topology changes
when the system performs a phase transition from hot de Sitter to
Schwarzschild-de Sitter and vice versa as it is allowed in this formalism,
since it is in a semiclassical approximation to quantum gravity, and in
quantum gravity topology changes of the psce can happen. In the $\Lambda
R^{2}=1$ case, i.e., the Nariai universe, one has that the possible
thermodynamic phases that appear as one increases the temperature from zero
are hot de Sitter only, hot de Sitter favored in relation to the Nariai
universe, hot de Sitter coexisting equally with the Nariai universe, the
Nariai universe favored in relation to hot de Sitter. Here there is no phase
with only the Nariai universe. There are topology changes when the system
performs a phase transition from hot de Sitter to the Nariai universe and vice
versa as it is allowed in this formalism. In the $\Lambda R^{2}>1$ case, phase
transitions between hot de Sitter and the Schwarzschild-de Sitter black hole,
can also be explored.

Thus, we have given a full thermodynamic description of the Schwarzschild-de
Sitter black hole space in a finite size cavity.

###### Acknowledgements.

We thank conversations with Tiago Fernandes and Francisco Gandum on the
Euclidean path integral approach to quantum gravity. We acknowledge financial
support from Fundação para a Ciência e Tecnologia - FCT through the project
No. UIDB/00099/2020.

##  Appendix A The Schwarzschild-de Sitter and Nariai spacetimes: Basics

###  A.1 The Schwarzschild-de Sitter spacetime

The line element of the Schwarzschild-de Sitter spacetime in spherical
coordinates $(t,r,\theta,\phi)$ is given by

$ds^{2}=-V(r)\,dt^{2}+\frac{dr^{2}}{V(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
(118)

where metric potential $V(r)$ has the form

$V(r)=1-\frac{2m}{r}-\frac{\Lambda r^{2}}{3},$ (119)

with $m$ being the spacetime mass and $\Lambda$ the cosmological constant
which we consider positive, $\Lambda>0$, see also Fig. A1. The coordinate
ranges are $-\infty<t<\infty$, $r_{+}<r<r_{\rm c}$, $0\leq\theta\leq\pi$, and
$0\leq\phi<2\pi$, where $r_{+}$ and $r_{\rm c}$ are the black hole and
cosmological horizons of the spacetime, respectively. The spacetime has
topology $R^{4}$. These coordinates can be further extended, e.g., through a
Kruskal-Szekeres extension, but it is not necessary here to do so.

The equation $V(r)=0$ is

$r-2m-\frac{\Lambda r^{3}}{3}=0,$ (120)

which can be written as $m=\frac{r}{2}\left(1-\frac{\Lambda r^{2}}{3}\right)$.
Note that Eq. (120) has at most two positive real roots. When it has roots,
one root corresponds to the black hole horizon $r_{+}$, with

$r_{+}=r_{+}(m,\Lambda)\,.$ (121)

The radius of the black hole horizon obeys the inequality $0\leq
r_{+}\leq\frac{1}{\sqrt{\Lambda}}$. The explicit form of $r_{+}(m,\Lambda),$
can be given since Eq. (120) is a cubic equation for $r$, but it is cumbersome
and there is no need to present it explicitly. Given $r_{+}$, then $V(r)$ in
Eq. (119) can be written as $V(r)=1-\frac{r_{+}\left(1-\frac{\Lambda
r_{+}^{2}}{3}\right)}{r}-\frac{\Lambda r^{2}}{3}$ or
$V(r)=1-\frac{r_{+}}{r}-\frac{\Lambda}{3r}\left(r^{3}-r_{+}^{3}\right)$, where
use of Eq. (120) has been made. The other root corresponds to the cosmological
horizon $r_{\rm c}$,

$r_{\rm c}=r_{\rm c}(m,\Lambda),$ (122)

with $r_{\rm c}\geq r_{+}$. The radius of the cosmological horizon obeys the
inequality $\frac{1}{\sqrt{\Lambda}}\leq r_{\rm
c}\leq\sqrt{\frac{3}{\Lambda}}$. The explicit form of $r_{\rm c}(m,\Lambda),$
can be given since Eq. (120) is a cubic equation for $r$, but it is cumbersome
and there is no need to present it explicitly.

Figure A1:  A drawing of the black hole with its event horizon $r_{+}$ and of
the cosmological horizon $r_{\rm c}$ in the Schwarzschild-de Sitter spacetime.
The topology of the 3-space is $R^{3}$.

On the other hand, since there are two roots of Eq. (120), $r_{+}$ and $r_{\rm
c}$, one can write

$V(r)=\frac{\Lambda}{3r}(r-r_{+})(r_{\rm c}-r)(r+r_{+}+r_{\rm c})\,,$ (123)

with

$r_{\rm c}^{2}+r_{\rm c}r_{+}+r_{+}^{2}=\frac{3}{\Lambda}\,,$ (124)

and

$r_{\rm c}r_{+}(r_{\rm c}+r_{+})=\frac{6m}{\Lambda}\,.$ (125)

Thus $\Lambda$ and $m$ can be swapped for $r_{+}$ and $r_{\rm c}$. Moreover,
Eq. (124) can be written as $r_{\rm c}^{2}+r_{\rm
c}r_{+}+r_{+}^{2}-\frac{3}{\Lambda}=0$ which is a quadratic either for $r_{\rm
c}$ in terms of $r_{+}$ or vice versa. The solution is

$r_{\rm c}=-\frac{r_{+}}{2}+\frac{r_{+}}{2}\sqrt{\frac{12-3\Lambda
r_{+}^{2}}{\Lambda r_{+}^{2}}}\,,$ (126)

or $r_{\rm c}=-\frac{r_{+}}{2}+\sqrt{\frac{3}{\Lambda}-\frac{3}{4}r_{+}^{2}}$.
Of course, the equation $r_{\rm c}^{2}+r_{\rm
c}r_{+}+r_{+}^{2}-\frac{3}{\Lambda}=0$ is also a quadratic for $r_{+}$ which
gives $r_{+}$ in terms of $r_{\rm c}$ in the same form of Eq. (126) with the
roles reversed. Another way of obtaining this is that since there are two
solutions of Eq. (120), $r_{+}$ and $r_{\rm c}$, one has from Eq. (120) that
$r_{\rm c}-2m-\frac{\Lambda r_{\rm c}^{3}}{3}=0$ and $r_{+}-2m-\frac{\Lambda
r_{+}^{3}}{3}=0$. Subtracting one equation from the other one eliminates $2m$
to get $(r_{\rm c}-r_{+})-\frac{\Lambda}{3}\left(r_{\rm
c}^{3}-r_{+}^{3}\right)=0$. Since $\left(r_{\rm
c}^{3}-r_{+}^{3}\right)=(r_{\rm c}-r_{+})(r_{\rm c}^{2}+r_{\rm
c}r_{+}+r_{+}^{2})$, one finds that $1-\frac{\Lambda}{3}(r_{\rm c}^{2}+r_{\rm
c}r_{+}+r_{+}^{2})=0$, i.e., $r_{\rm c}^{2}+r_{\rm
c}r_{+}+r_{+}^{2}-\frac{3}{\Lambda}=0$. One can then write the solution of
$r_{\rm c}$ in term of $r_{+}$ as $r_{\rm
c}=-\frac{r_{+}}{2}+\sqrt{\frac{3}{\Lambda}-\frac{3}{4}r_{+}^{2}}$, which is
Eq. (126). In addition, Eq. (125) with the help of Eq. (124) can be written as
$r_{\rm c}^{2}+r_{\rm c}r_{+}-\frac{2mr_{+}^{2}}{r_{+}-2m}=0$, which is a
quadratic either for $r_{\rm c}$ in terms of $r_{+}$ or vice versa. The
solution is

$r_{\rm
c}=-\frac{r_{+}}{2}+\frac{r_{+}}{2}\sqrt{\frac{r_{+}+6m}{r_{+}-2m}}\,,$ (127)

or $r_{\rm c}=\frac{r_{+}}{2}\left(\sqrt{\frac{r_{+}+6m}{r_{+}-2m}}-1\right)$.
Another way of obtaining this is that since there are two solutions of Eq.
(120), $r_{+}$ and $r_{\rm c}$, one has from Eq. (120) that $r_{\rm
c}r_{+}^{3}-2mr_{+}^{3}-\frac{\Lambda r_{\rm c}^{3}r_{+}^{3}}{3}=0$ and
$r_{+}r_{\rm c}^{3}-2mr_{\rm c}^{3}-\frac{\Lambda r_{+}^{3}r_{\rm
c}^{3}}{3}=0$. Subtracting one equation from the other one eliminates
$\Lambda$ to get $-r_{+}r_{\rm c}(r_{\rm c}^{2}-r_{+}^{2})+2m(r_{\rm
c}^{3}-r_{+}^{3})=0$. Thus, $-r_{+}r_{\rm c}(r_{\rm c}-r_{+})(r_{\rm
c}+r_{+})+2m(r_{\rm c}-r_{+})(r_{\rm c}^{2}+r_{\rm c}r_{+}+r_{+}^{2})=0$,
i.e., $-r_{+}r_{\rm c}(r_{\rm c}+r_{+})+2m(r_{\rm c}^{2}+r_{\rm
c}r_{+}+r_{+}^{2})=0$. So, $r_{\rm c}^{2}+r_{\rm
c}r_{+}-\frac{2mr_{+}^{2}}{r_{+}-2m}=0$, with solution $r_{\rm
c}=-\frac{r_{+}}{2}+\frac{r_{+}}{2}\sqrt{\frac{r_{+}+6m}{r_{+}-2m}}$ which is
Eq. (127).

Note that $r_{\rm c}=r_{+}$ for $r_{\rm c}=r_{+}=\frac{1}{\sqrt{\Lambda}}=3m$,
and this happens when $9m^{2}\Lambda=1$. In this limit one can have either an
extremal Schwarzschild-de Sitter spacetime, where the two regions off the
extremal horizon, the inside and the outside regions, are time dependent, or
the Nariai spacetime where the topology of the space changes, see below. For
$9m^{2}\Lambda>1$ there are no horizons, the spacetime is asymptotically de
Sitter, with a massive naked singularity at the center.

###  A.2 The Nariai spacetime

The line element of the Nariai spacetime in spherical coordinates
$(t,{z},\theta,\phi)$, is given by

$ds^{2}=-V({z})\,dt^{2}+\frac{d{z}^{2}}{V({z})}+\frac{1}{\Lambda}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right),$
(128)

where the metric potential $V({z})$ has the form

$V({z})=1-\Lambda{z}^{2},$ (129)

with $\Lambda$ being the cosmological constant which we consider positive,
$\Lambda>0$, see also Fig. A2. The coordinate ranges are $-\infty<t<\infty$,
${z}_{+}<{z}<{z}_{\rm c}$, where ${z}_{+}$ and ${z}_{\rm c}$ are the horizons
of the spacetime, $0\leq\theta\leq\pi$, and $0\leq\phi<2\pi$. Note that the
spacetime has topology $R^{2}\times S^{2}$. These coordinates can be further
extended, e.g., through a Kruskal-Szekeres extension, but it is not necessary
here to do so.

Figure A2:  A drawing of the Nariai universe with its event horizon $z_{+}$
and its cosmological horizon $z_{\rm c}$. The cylindrical character of the
Nariai space is made clear after identifying the two vertical end lines of the
diagram. The radius $R$ is the radius of the cylinder. The labelling of the
two horizons $z_{+}$ and $z_{\rm c}$ is convention as they are of the same
type. The topology of the 3-space is $R\times S^{2}$.

The equation $V({z})=0$ is

$1=\Lambda{z}^{2}.$ (130)

In general Eq. (130) has two roots. One root corresponds to the black hole
horizon ${z}_{+}$, by convention, with

${z}_{+}=-\frac{1}{\sqrt{\Lambda}}\,.$ (131)

The other root corresponds to the cosmological horizon ${z}_{\rm c}$, by
convention, with

${z}_{\rm c}=+\frac{1}{\sqrt{\Lambda}},$ (132)

with ${z}_{\rm c}\geq{z}_{+}$.

Note that when

$\Lambda=\infty\,,$ (133)

both roots coincide,

${z}_{+}={z}_{\rm c}=0,$ (134)

and in this extremal limit the spacetime disappears.

Note also that when $\Lambda=0$, the $z_{+}$ and $z_{\rm c}$ have roots

$z_{+}=-\infty\,,\quad z_{\rm c}=+\infty\,,$ (135)

and in this limit there are no horizons, one is in the presence of simply a
Minkowski space with two coordinates possibly wrapped around, i.e., one can
have a torus, a cylinder, or an infinite plane. This can be seen from the
angular part of Eq. (128) by doing $\theta=\sqrt{\Lambda}\,x$,
$\phi=\sqrt{\Lambda}\,y$, and then doing $\Lambda\to 0$ to give
$ds^{2}=-dt^{2}+dz^{2}+dx^{2}+dy^{2}$, where the range of the coordinates $x$
and $y$ is $0\leq x\leq x_{1}$ and $0\leq y\leq y_{1}$, respectively, with
$x_{1}$ and $y_{1}$ having any value one chooses from a finite value to
infinite.

##  Appendix B The Nariai limit from the Schwarzschild-de Sitter space in a
thermodynamic setting

In order to understand the limiting thermodynamic process of obtaining a
Nariai space from a Schwarzschild-de Sitter space in the limit $\Lambda
R^{2}=1$, $\frac{r_{+}}{R}=1$, and $\frac{r_{\rm c}}{R}=1$, it is useful to
resort to Fig. B1.

To be complete we give again the Schwarzschild-de Sitter Euclidean line
element

$\displaystyle ds^{2}=V(r)\,dt^{2}+$
$\displaystyle\frac{dr^{2}}{V(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
$\displaystyle 0\leq t<\beta^{\rm H}_{+}\,,\,r_{+}\leq r\leq R\,,$ (136)

where the metric potential $V(r)$ has the form

$V(r)=1-\frac{2m}{r}-\frac{\Lambda r^{2}}{3}\,.$ (137)

Note that the range of coordinates of Euclidean time $t$ is $0\leq
t<\beta^{\rm H}_{+}$, where $\beta^{\rm H}_{+}$ is the period of the
coordinate such that the line element given by Eqs. (136) and (137) has no
conical singularities. The relation with the Hawking temperature $T^{\rm
H}_{+}$ is $\beta^{\rm H}_{+}=\frac{1}{T^{\rm H}_{+}}$. In addition, due to
the reservoir at radius $R$ the range of coordinates of $r$ is now $r_{+}\leq
r\leq R$. Note now that this is Euclidean space and the topology of this space
is $R^{2}\times S^{2}$. There are two horizon for $V(r)=0$, see Eq. (137), the
black hole horizon $r_{+}$ and the cosmological horizon $r_{\rm c}$.

The local Tolman temperature $T$ of the heat reservoir at $R$ has to be
treated with care. It is

$T=\frac{T^{\rm H}_{+}}{\sqrt{V(R)}}\,,$ (138)

with $T^{\rm H}_{+}$ being the Hawking temperature given by $T^{\rm
H}_{+}=\frac{\kappa_{+}}{2\pi}$, and $\kappa_{+}$ being the surface gravity of
the black hole horizon and $V(R)$ is the potential at $R$. For the line
element of Eq. (136) one has $\kappa_{+}=\frac{1}{2}{V^{\prime}(r_{+})}$,
where a prime means derivative with respect to $r$, and so the Hawking
temperature for the black hole is $T^{\rm
H}_{+}=\frac{1}{4\pi}\left(\frac{dV}{dr}\right)_{r_{+}}$. Using Eq. (137) we
get

$T^{\rm H}_{+}=\frac{1}{4\pi r_{+}}\left(1-\Lambda r_{+}^{2}\right).$ (139)

The potential at $R$ is

$V(R)=1-\frac{2m}{R}-\frac{\Lambda R^{2}}{3}\,.$ (140)

Figure B1:  Drawings of a black hole horizon inside a heat reservoir in the
Schwarzschild-de Sitter black hole space on the left, and of the Nariai
universe with one of its horizons inside a heat reservoir on the right.
Specifically, on the left the drawing shows a slant view of a $t={\rm
constant}$ and $\theta={\rm constant}$ space of the black hole horizon $r_{+}$
inside a heat reservoir at temperature $T$ and radius $R$ in the
Schwarzschild-de Sitter geometry. Outside $R$ there is the cosmological
horizon $r_{\rm c}$. The Euclideanized space and its boundary have
$R^{2}\times S^{2}$ and $S^{1}\times S^{2}$ topologies, respectively, with the
$S^{1}$ subspace having proper length $\beta=\frac{1}{T}$. On the right, the
drawing shows a slant view of a $t={\rm constant}$ and $\theta={\rm constant}$
space of the horizon $z_{+}$ inside a heat reservoir at temperature $T$, with
cylindrical radius $R$, and situated at $Z$ in the Nariai universe geometry.
Outside $Z$ there is the cosmological horizon $z_{\rm c}$. The Euclideanized
space and its boundary have $R^{2}\times S^{2}$ and $S^{1}\times S^{2}$
topologies, respectively, with the $S^{1}$ subspace having proper length
$\beta=\frac{1}{T}$. In pictorial terms it is clear how the Schwarzschild-de
Sitter black hole space originates the Nariai universe, with the slant view of
the Schwarzschild-de Sitter space helping in the visualization of the process.
Indeed, if the two Schwarzschild-de Sitter different horizon radii, $r_{+}$
and $r_{\rm c}$, are squeezed into the heat reservoir radius $R$, then the two
horizons pinch off to form the Nariai universe with the heat reservoir still
at temperature $T$, with cylindrical radius $R$, and situated now at some $Z$
in-between the two displaced distinct horizons $z_{+}$ and $z_{\rm c}$. See
text for more details.

The Nariai solution can be found from the Schwarzschild-de Sitter solution in
the limit that the two horizons $r_{+}$ and $r_{\rm c}$ coincide. Here, we
have a heat reservoir at $R$ that acts for the inside region which is a cavity
with the black hole. This heat reservoir at $R$ is in-between $r_{+}$ and
$r_{\rm c}$, and thus the limit is such that $r_{+}$, $R$, and $r_{\rm c}$
coincide. Now, if we do $r_{+}\to R$ then Eq. (138), $T=\frac{T^{\rm
H}_{+}}{\sqrt{V(R)}}$, gives at face value that the heat reservoir is at very
high temperature since $V(R)\to 0$. But, there is a way to have $T$ finite
with $r_{+}\to R$, and this is to do concomitantly $T^{\rm H}_{+}\to 0$. Since
$T^{\rm H}_{+}=\frac{1}{4\pi r_{+}}\left(1-\Lambda r_{+}^{2}\right)$, see Eq.
(139), $T^{\rm H}_{+}\to 0$ means $1-\Lambda r_{+}^{2}\to 0$. Since $r_{+}\to
R$, we put,

$\frac{r_{+}}{R}=1-\varepsilon\,,$ (141)

$\varepsilon\ll 1$, which implies

$\sqrt{\Lambda R^{2}}=1-\delta\,,$ (142)

where $\delta\equiv\frac{1}{2}(1-\Lambda r_{+}^{2})-\varepsilon$, $\delta\ll
1$, and $\frac{\varepsilon}{\delta}$ is of order one. Thus, $R$ and the length
scale $\frac{1}{\sqrt{\Lambda}}$ are equal at zeroth order. Since $r_{+}\to
R\to\frac{1}{\sqrt{\Lambda}}$ then from Eq. (5) one has $\frac{r_{\rm
c}}{R}\to 1$, i.e., one has $r_{\rm c}\to\frac{1}{\sqrt{\Lambda}}$, and since
$\frac{1}{\sqrt{\Lambda}}\to R$, then $r_{\rm c}\to R$. From the expansion of
Eqs. (141) and (142) one has that Eq. (139) gives

$T^{\rm H}_{+}=\frac{\delta+\varepsilon}{2\pi R}\,,$ (143)

in first order, as required. To understand the behavior of $V(r)$ near $R$ in
this limit we use Eq. (140). It gives

$V(R)=\varepsilon\left(\varepsilon+2\delta\right),$ (144)

in first order. Thus, Eq. (138), i.e., $T=\frac{T^{\rm H}_{+}}{\sqrt{V(R)}}$
gives

$T=\frac{1}{2\pi
R}\frac{\frac{\varepsilon}{\delta}+1}{\sqrt{\frac{\varepsilon}{\delta}\left(\frac{\varepsilon}{\delta}+2\right)}}\,,$
(145)

which given a $RT$, i.e., given a $T$, implies some
$\frac{\varepsilon}{\delta}$, and so is finite and consistent.

But we have not finished. The metric potential $V(R)$ given in Eq. (136)
vanishes in these coordinates, and the line element of Eq. (136) looses sense.
So, we have to pay attention to this limit indeed with care. It is the Nariai
limit with a reservoir $R$ in the middle, see Fig. (B1), an interesting case.
Expanding the metric potential $V(r)$ near $r_{+}$ in a Taylor series gives
$V(r)=\left(\frac{dV}{dr}\right)_{r=r_{+}}\left(r-r_{+}\right)+\frac{1}{2}\left(\frac{d^{2}V}{dr^{2}}\right)_{r=r_{+}}\left(r-r_{+}\right)^{2}$
plus higher order terms. Recall that
$\left(\frac{dV}{dr}\right)_{r=r_{+}}=4\pi T^{\rm H}_{+}$ and find
$\left(\frac{d^{2}V}{dr^{2}}\right)_{r=r_{+}}=-\frac{2}{r_{+}^{2}}$. Now, in
this limit $r_{+}=R$, so
$\left(\frac{d^{2}V}{dr^{2}}\right)_{r=r_{+}}=-\frac{2}{R^{2}}$. So,

$V(r)=4\pi T^{\rm H}_{+}(r-r_{+})-\frac{1}{R^{2}}\left(r-r_{+}\right)^{2}\,,$
(146)

plus higher order terms. Make the coordinate transformations
$(t,r)\to(\bar{t},{\bar{z}})$ as

$r-r_{+}=4\pi T^{\rm
H}_{+}R^{2}\sin^{2}\left(\frac{1}{2}\frac{{\bar{z}}}{R}\right)\,,\quad\quad
t=\frac{\bar{t}}{2\pi T^{\rm H}_{+}R}\,,$ (147)

with $0\leq t\leq\frac{1}{T^{\rm H}_{+}}$ corresponding to $0\leq{\bar{t}}\leq
2\pi R$, and $r_{+}\leq r\leq R$ corresponding to $0\leq\bar{z}\leq\bar{Z}$.
Then obtain from Eq. (137) that

$V(r)=V(r-r_{+})=V(\bar{z})=(2\pi T^{\rm
H}_{+}R)^{2}\sin^{2}\left(\frac{{\bar{z}}}{R}\right)\,,$ (148)

and from Eq. (136) obtain the line element,

$ds^{2}=\sin^{2}\left(\frac{{\bar{z}}}{R}\right)\,d{\bar{t}}^{2}+d{\bar{z}}^{2}+R^{2}d\Omega^{2}\,,$
(149)

which is a form of the Nariai line element. Note now from Eq. (148) that
$V(R)=V(R-r_{+})=V(\bar{Z})=(2\pi T^{\rm
H}_{+}R)^{2}\sin^{2}\left(\frac{{\bar{Z}}}{R}\right)$, and so from Eq. (138),
i.e., $T=\frac{T^{\rm H}_{+}}{\sqrt{V(R)}}$, one has

$T=\frac{1}{2\pi R\sin\left(\frac{{\bar{Z}}}{R}\right)}\,.$ (150)

This means that the ensemble boundary values $T$ and $R$ specify automatically
the maximum value for ${\bar{z}}$, namely ${\bar{Z}}$. So, for the ensemble
with boundary data one has $0<t<2\pi R$, $0\leq{\bar{z}}\leq{\bar{Z}}$,
$0\leq\theta\leq\pi$, $0\leq\phi<2\pi$. In turn $0\leq{\bar{Z}}\leq R\pi$.
Note that ${\bar{z}}=0$ and ${\bar{z}}=R\pi$ are horizons.

We can continue further. Let us make another coordinate transformation,
$z=R\cos\frac{\bar{z}}{R}$, Then

$ds^{2}=V(z)\,dt^{2}+\dfrac{dz^{2}}{V(z)}+R^{2}d\Omega^{2}\,,$ (151)

where we have dropped the bar in $\bar{t}$. Then, the metric potential $V(z)$
is

$V=1-\frac{z^{2}}{R^{2}}\,,$ (152)

the one given in Eq. (129). So, for the ensemble with boundary data one has
$0<t<2\pi R$, $-R<z<Z$, $0\leq\theta\leq\pi$, $0\leq\phi<2\pi$. In turn
$-R\leq Z\leq R$. Note that $z=-R$ and $z=R$ are horizons. The line element
given in Eq. (151) and (152) corresponds to a two-dimensional de Sitter space
times a sphere, the topology is $R^{2}\times S^{2}$. It has two horizons, we
label $z_{+}=-R$ and $z_{\rm c}=R$, but now they are just labels, since the
two horizons have the same character.

We have that the heat reservoir temperature $T$ is now

$T=\frac{T^{\rm H}_{\rm+\,Nariai}}{\sqrt{V(Z)}}\,,$ (153)

with $T^{\rm H}_{\rm+\,Nariai}$ being the Hawking temperature given by $T^{\rm
H}_{\rm+\,Nariai}=\frac{\kappa_{+}}{2\pi}$, and $\kappa_{+}$ being the surface
gravity of the black hole horizon. For the metric given in Eq. (151) one has
$\kappa_{+}=\frac{1}{2}{V^{\prime}(z_{+})}$, and so the Hawking temperature
for + horizon is $T^{\rm
H}_{\rm+\,Nariai}=\frac{1}{4\pi}\left(\frac{dV}{dz}\right)_{z_{+}}$. Using
Eqs. (151) and (152) we get

$T^{\rm H}_{\rm+\,Nariai}=\frac{1}{2\pi R}.$ (154)

So, from Eqs. (153) and (154) one finds

$T=\frac{1}{2\pi R\sqrt{1-\frac{Z^{2}}{R^{2}}}}\,,$ (155)

where $Z$ is the boundary of $Z$. For a given $T$ at the boundary, for the
ensemble, there are two solutions of Eq. (155) in general, namely, one for $Z$
between $-R$ and 0 and the other for $Z$ between 0 and $R$. These two
solutions may be thought of yielding different physical situations. Note also
that in Schwarzschild-de Sitter the two solutions were for the horizon
$r_{+}$, one small $r_{+1}$ the other large $r_{+2}$, here the two solutions
are not for the horizons but instead for the boundary $Z$, one $Z_{1}$, the
other $Z_{2}$, with

$Z_{1}=-R\sqrt{1-\frac{1}{(2\pi RT)^{2}}}\,,$ (156)

and

$Z_{2}=R\sqrt{1-\frac{1}{(2\pi RT)^{2}}}\,,$ (157)

We are free to choose where to put the boundary, we have two choices, either
$Z_{1}$ or $Z_{2}$, noting that $z_{+}$ and $z_{\rm c}$ can also be exchanged
if one wants. Exchanging $Z_{1}$ with $Z_{2}$ and concomitantly exchanging
$z_{+}$ with $z_{\rm c}$ reverses to the original situation. The boundary is
at $Z_{1}$ or $Z_{2}$, has cylindrical radius $R$, and acts as a reservoir for
the inner region, i.e., $z\leq Z_{1}$ or $z\leq Z_{2}$ depending on where we
choose to put the boundary. Note, in addition that the reservoir instead of
being two-dimensional with space topology $S^{2}$ as in the de Sitter space,
it is still two dimensional but now with topology $R\times S^{1}$, i.e., it is
a cylinder.

From Eqs. (156) and (157) we see that for

$RT<\frac{1}{2\pi}\,,$ (158)

there are no solutions for $Z_{1}$ or $Z_{2}$, so in this case the boundary in
$Z$ does not exist, and so there is no Nariai solution for the thermodynamic
problem in the canonical ensemble. The reason is clear if one thinks in
thermal wavelengths, indeed when the temperatures are relatively very small,
the associated thermal wavelengths are very long and there is no boundary Z
that can accommodate them.

From Eqs. (156) and (157) we see that for

$RT\geq\frac{1}{2\pi}\,,$ (159)

there are two Nariai solutions, one with one horizon $z=-R$ and boundary
$Z_{1}$, the other with with one horizon $z=-R$ and boundary $Z_{2}$, both
boundaries can be picked up. When the equality sign holds there is one
solution with $Z_{1}=Z_{2}=0$, so in this case the boundary in $Z$ pops up in
the middle, at $z=0$, and so $-R\leq z\leq 0$. The reservoir is at
$Z_{1}=Z_{2}=0$ and has cylindrical radius $R$ and the horizon is at $z=-R$.

Three comments are in order. The first comment is that the resulting metric
inside the reservoir is in this case described by the Nariai universe as we
have seen. The procedure of obtaining the Nariai metric as a limit from the
Schwarzschild-de Sitter metric is known and has different versions, e.g., see
gp ; diaslemoscardosonariaiinhigherd . However, we have done it here in a
completely different manner and in a completely different context that is
related naturally to thermodynamics. The procedure we used is similar to that
described in prd97  for the extremal limit of nonextremal electrically charged
black holes. Here, in Eq. (146), the potential expansion takes the form
$V(r)=4\pi T^{\rm H}_{+}(r-r_{+})-\frac{1}{R^{2}}\left(r-r_{+}\right)^{2}$
instead of $V(r)=4\pi T^{\rm
H}_{+}(r-r_{+})+\frac{1}{R^{2}}\left(r-r_{+}\right)^{2}$, i.e., a plus instead
of the minus sign of prd97  appears in the second term of the expansion. As a
result, we have arrived at the Nariai metric instead of obtaining the
Bertotti-Robinson metric as in prd97 . The second comment, is that in the
above considerations, we have discussed two completely different limits, one
is the high-temperature limit and the other is the extremal Nariai limit.
However, they can be combined if the boundary $Z_{1}\to-R$. The third comment
is that we have done the Nariai limit from below, i.e., from $\Lambda
R^{2}<1$. If we do the limit $\Lambda R^{2}\to 1$ from above, i.e., from
$\Lambda R^{2}>1$, we get the same result, namely, the Nariai universe inside
a heat reservoir in the canonical ensemble.

##  Appendix C Deduction of formulas of Sec. VI

Here, we deduce in detail some equations found in Sec. VI. Specifically, we
want to deduce Eqs. (96), (100), (108), and (112), and study Eqs. (116) and
(117).

For the derivation of Eqs. (96), (100), (108), and (112) it is convenient to
shorten the notation and define the variables $\sqrt{\Lambda R^{2}}$ and
$\frac{r_{+}}{R}$ as $x$ and $y$ variables, namely,

$x\equiv\sqrt{\Lambda R^{2}}\,,$ (160) $y\equiv\frac{r_{+}}{R}\,.$ (161)

With these definitions we want to find Eqs. (96) and (100) of the $x^{2}<1$
case, and Eqs. (108) and (112) of the $x^{2}>1$ case.

$x^{2}{\mathbf{<}1}$:

Let us deduce Eq. (96). We have to find the behavior of $y_{2}$, i.e.,
$r_{+2}$, at $x^{2}$ near 1, i.e., $\Lambda R^{2}$ near 1. From Eq. (85), or
from Eq. (92), one finds that there is a solution $y_{2}(x)$ for each $x$. The
maximum of the curve is at $x=1$ and $y_{2}=1$, and the derivative there is
not well defined. So, it is important to discuss the behavior and the
properties of $y_{2}$ when $x\to 1$. For that we define infinitesimal
quantities $\delta$ and $\varepsilon$, related to $x$ and $y$, respectively,
by

$x=1-\delta\,,\quad\quad\quad y=1-\varepsilon\,,$ (162)

with $\delta\ll 1$ and $\varepsilon\ll 1$, both positive, and with
$\frac{\varepsilon}{\delta}$ being a quantity of order one. Indeed, $\delta>0$
for $x<1$, and $\varepsilon\geq 0$ always. Then, from Eq. (85) one gets

$w=\frac{2(\frac{\varepsilon}{\delta}+1)}{\sqrt{\frac{\varepsilon}{\delta}(\frac{\varepsilon}{\delta}+2)}}\,,$
(163)

valid in first order, in the order we are working. One sees that for $w$
finite $\frac{\varepsilon}{\delta}$ is finite, so generically here in this
calculation $\varepsilon$ is indeed of the order $\delta$. From Eq. (86) one
also gets

$\frac{dy}{dx}=\frac{\varepsilon}{\delta}\,,$ (164)

also valid in the order we are working. We obtain from Eq. (163) that
$\frac{\varepsilon}{\delta}=\frac{w}{\sqrt{w^{2}-4}}-1$ with $w\geq 2$. Thus,
since here $\frac{dy}{dx}=\frac{\varepsilon}{\delta}$, Eq. (164), one finds
that

$\frac{dy}{dx}=\frac{w}{\sqrt{w^{2}-4}}-1\,,\quad\quad w\geq 2\,,$ (165)

which is Eq. (96), the equation we wanted to deduce.

Let us deduce now Eq. (100). We want to find $y_{1}=y_{2}$, the coincident
solution for fixed $x$, and $w$ varying. This is when $\frac{dw}{dy}=0$, see
Eq. (89), i.e., ${S}(x,y)=0$. From Eq. (90) we have then that the bifurcation
points where $y_{1}=y_{2}$ for $x={\rm constant}$ are given by Eq. (99). We
are interested in finding $y_{1}=y_{2}$ when $x\to 1$. Instead of working with
Eq. (99) we work with Eq. (90) which is more direct. We find from $x=1-\delta$
of Eq. (162), that Eq. (90) in this limit is
${S}=(y-1)^{3}(y^{2}+3y+4)-4\delta(y-1)^{2}(y+1)(1+y+y^{2})+4\delta^{2}(2y^{2}+y^{5})$.
In addition, from $y=1-\varepsilon$ of Eq. (162), Eq. (90) becomes then
${S}=12\delta^{2}-24\delta\varepsilon^{2}-8\varepsilon^{3}$. As we have seen,
the coincident root is given when $\frac{dw}{dy}=0$, i.e., ${S}=0$. Then,
$12\delta^{2}-24\delta\varepsilon^{2}-8\varepsilon^{3}=0$, i.e.,
$\delta^{2}-2\delta\varepsilon^{2}-\frac{2}{3}\varepsilon^{3}=0$. This is a
quadratic in $\delta$, with solution given by
$\delta=\sqrt{\frac{2}{3}\varepsilon^{3}+\varepsilon^{4}}+\varepsilon^{2}$,
i.e.
$\delta=\varepsilon^{\frac{3}{2}}\sqrt{\frac{2}{3}+\varepsilon}+\varepsilon^{2}$.
Since $\varepsilon\ll 1$ the dominant term gives

$\delta=\left(\frac{2}{3}\varepsilon^{3}\right)^{\frac{1}{2}}\,.$ (166)

Thus, $\delta$ goes with $\varepsilon^{\frac{3}{2}}$ or, if one prefers,
$\varepsilon$ goes with $\delta^{\frac{2}{3}}$, indeed
$\varepsilon=\left(\frac{3}{2}\delta^{2}\right)^{\frac{1}{3}}$. Recall that
$x=1-\delta$, and so $x^{2}=1-2\delta$ in this approximation, and that
$y=1-\varepsilon$. Then, substituting $\delta$ for $x^{2}$ and $\varepsilon$
for $y$ in Eq. (166), one has

$x^{2}=1-\left(\frac{8}{3}(1-y)^{3}\right)^{\frac{1}{2}}\,.$ (167)

We see that Eq. (167) is Eq. (99) in the limit $x\to 1$ and $y\to 1$.
Inverting Eq. (167) one has that the coincident solution takes the form

$y_{1}=y_{2}=1-\left(\frac{3}{8}\left(1-x^{2}\right)^{2}\right)^{\frac{1}{3}}\,,$
(168)

valid for $1-x^{2}\ll 1$, which is Eq. (100), the equation we wanted to
deduce.

$x^{2}{\mathbf{>}1}$:

Let us deduce Eq. (108). We have to find $y_{2}$, i.e., $r_{+2}$. From Eq.
(85), or from Eq. (106), one finds that there is a solution $y_{2}(x)$ for
each $x$. The maximum of the curve is at $x=1$ and $y_{2}=1$, and the
derivative there it is not well defined. So, it is important to discuss the
behavior and the properties of $y_{2}$ when $x\to 1$ from above. For that we
define infinitesimal quantities $\bar{\delta}$ and $\bar{\varepsilon}$,
related to $x$ and $y$, respectively, by

$x=1+{\bar{\delta}}\,,\quad\quad\quad y=1-{\bar{\varepsilon}}\,,$ (169)

with $\bar{\delta}\ll 1$ and $\bar{\varepsilon}\ll 1$, both positive, and with
$\frac{\bar{\varepsilon}}{\bar{\delta}}$ being a quantity of order one.
Indeed, $\bar{\delta}>0$ for $x>1$, and $\bar{\varepsilon}>0$ always. Then,
from Eq. (85) one gets

$w=\frac{2(\frac{{\bar{\varepsilon}}}{{\bar{\delta}}}-1)}{\sqrt{\frac{{\bar{\varepsilon}}}{{\bar{\delta}}}(\frac{{\bar{\varepsilon}}}{{\bar{\delta}}}-2)}}\,,$
(170)

valid in first order, in the order we are working. One sees that for $w$
finite $\frac{\bar{\varepsilon}}{\bar{\delta}}$ is finite, so generically here
in this calculation $\bar{\varepsilon}$ is of the order of $\bar{\delta}$.
From Eq. (86) one also gets

$\frac{dy}{dx}=-\frac{{\bar{\varepsilon}}}{{\bar{\delta}}}\,,$ (171)

also valid in the order we are working. Since ${\bar{\delta}}>0$ and
${\bar{\varepsilon}}>0$, we obtain from Eq. (170)
$\frac{{\bar{\varepsilon}}}{{\bar{\delta}}}=\frac{w}{\sqrt{w^{2}-4}}+1$ with
$w\geq 2$. Thus, we have from Eq. (171) that

$\frac{dy}{dx}=-\frac{w}{\sqrt{w^{2}-4}}-1\,,\quad\quad w\geq 2\,,$ (172)

which is Eq. (108), the equation we wanted to deduce.

Let us deduce now Eq. (112). We want to find $y_{1}=y_{2}$, the coincident
solution for fixed $x$, and $w$ varying. This is when $\frac{dw}{dy}=0$, see
Eq. (89), i.e., ${S}(x,y)=0$. From Eq. (90) we have then that the bifurcation
points where $y_{1}=y_{2}$ for $x={\rm constant}$ are given by Eq. (111). We
are interested in finding $y_{1}=y_{2}$ when $x\to 1$ from above. Instead of
working with Eq. (111) we work with Eq. (90) which is more direct. We find
Then, from Eq. (105) one has $y_{\rm e}=1-2{\bar{\delta}}$, plus higher order.
For that, let $x=1+{\bar{\delta}}$ as in Eq. (169). Then, from Eq. (105) one
has $y_{\rm e}=1-2{\bar{\delta}}$, plus higher order terms. Now we have to
work out the vicinity of $y_{\rm e}$. Then put $y=y_{\rm e}-{\epsilon}$ where
again $y_{\rm e}$ is the root of the equation $y^{2}+y+1-\frac{3}{x^{2}}=0$,
with solution given by Eq. (105), i.e., $y_{\rm
e}=-\frac{1}{2}+\sqrt{3}\sqrt{\frac{1}{x^{2}}-\frac{1}{4}}$, and note that
$\epsilon$ and $\bar{\varepsilon}$ are different quantities. Then, these two
equations yield $y=1-2{\bar{\delta}}-{\epsilon}$. From $x=1+{\bar{\delta}}$ of
Eq. (169) we find that Eq. (90) is
${S}=(y-1)^{3}(y^{2}+3y+4)+4{\bar{\delta}}(y-1)^{2}(y+1)(1+y+y^{2})+(2y^{2}+y^{5})4{\bar{\delta}}^{2}$.
From $y=1-2{\bar{\delta}}-{\epsilon}$ we have just found, we find that Eq.
(90) is now
${S}=12{\bar{\delta}}^{2}+24{\bar{\delta}}{\epsilon}^{2}-8{\epsilon}^{3}$. The
coincident root is given when $\frac{\partial w}{\partial y}=0$, i.e.,
${S}=0$. Then,
$12{\bar{\delta}}^{2}+24{\bar{\delta}}{\epsilon}^{2}-8{\epsilon}^{3}$, i.e.,
${\bar{\delta}}^{2}+2{\bar{\delta}}{\epsilon}^{2}-\frac{2}{3}{\epsilon}^{3}=0$.
The solution is
${\bar{\delta}}=-{\epsilon}^{2}+\sqrt{{\epsilon}^{4}+\frac{2}{3}{\epsilon}^{3}}$.
It turns out that equation $\frac{\partial w}{\partial y}=0$ gives a self-
consistent solution if ${\bar{\delta}}\ll{\bar{\epsilon}}$, see below. Since
${\epsilon}\ll 1$ the dominant term gives
${\bar{\delta}}=(\frac{2}{3}{\epsilon}^{3})^{\frac{1}{2}}$. We defined
${\bar{\varepsilon}}$ as $y=1-{\bar{\varepsilon}}$ and also had
$y=1-2{\bar{\delta}}-{\epsilon}$, so
${\bar{\varepsilon}}=2{\bar{\delta}}+{\epsilon}$. But ${\bar{\delta}}$ goes
with ${\epsilon}^{\frac{3}{2}}$, so in this order
${\bar{\varepsilon}}={\epsilon}$, so that

${\bar{\delta}}=\left(\frac{2}{3}{\bar{\varepsilon}}^{3}\right)^{\frac{1}{2}}\,.$
(173)

Thus, $\bar{\delta}$ goes with $\bar{\varepsilon}^{\frac{3}{2}}$ or, if one
prefers, $\bar{\varepsilon}$ goes with $\bar{\delta}^{\frac{2}{3}}$, indeed
$\bar{\varepsilon}=\left(\frac{3}{2}\bar{\delta}^{2}\right)^{\frac{1}{3}}$.
Recall that $x=1+\bar{\delta}$, and so $x^{2}=1-2\bar{\delta}$ in this
approximation, and that $y=1-\bar{\varepsilon}$. Then, substituting
$\bar{\delta}$ for $x^{2}$ and $\bar{\varepsilon}$ for $y$ in Eq. (173), one
has

$x^{2}=1+\left(\frac{8}{3}(1-y)^{3}\right)^{\frac{1}{2}}\,.$ (174)

We see that Eq. (174) is Eq. (111) in the limit $x\to 1$ from above and $y\to
1$. Inverting Eq. (174) one has that the coincident solution takes the form

$y_{1}=y_{2}=1-\left(\frac{3}{8}\left(x^{2}-1\right)^{2}\right)^{\frac{1}{3}}\,,$
(175)

valid for $x^{2}-1\ll 1$, which is Eq. (112), the equation we wanted to
deduce.

Finally, we calculate the implications of Eqs. (116) and (117). We have
defined $u$ as

$u\equiv\frac{r_{\rm c}}{R}\,,$ (176)

as the cosmological radius in units of $R$, which from Eq. (126) means that
$u$ obeys

$u=-\frac{y}{2}+\sqrt{\frac{3}{x^{2}}-\frac{3}{4}y^{2}}\,.$ (177)

Calculating $\frac{du}{dy}$ at the neighborhood of the point $x=1$, $y=1$,
$u=1$, and recalling that we are interested in the large black hole, i.e., the
one with subscript 2, one finds from Eq. (177), with the help of Eqs. (164)
and (171) , that

$\left(\frac{du_{2}}{dx}\right)_{x=1_{-}}=\left(\frac{dy_{2}}{dx}\right)_{x=1_{+}}$
(178)

and

$\left(\frac{du_{2}}{dx}\right)_{x=1_{+}}=\left(\frac{dy_{2}}{dx}\right)_{x=1_{-}}$
(179)

where $1_{-}$ and $1_{+}$ means that one is taking the limit $x\to 1$ from
below, i.e., $x<1$, and from above, i.e., $x>1$, respectively. We can now
deduce some further specific properties of the cosmological horizon. In Eq.
(172) we found
$\left(\frac{dy_{2}}{dx}\right)_{x=1_{+}}=-\frac{w}{\sqrt{w^{2}-4}}-1$ so that
from Eq. (178) we have
$\left(\frac{du_{2}}{dx}\right)_{x=1_{-}}=-\frac{w}{\sqrt{w^{2}-4}}-1$. Thus,
$\left(\frac{du_{2}}{dx}\right)_{x=1_{-}}\to-\infty$ for $w\to 2$, and
$\left(\frac{du_{2}}{dx}\right)_{x=1_{+}}\to-2$ for $w\to\infty$. In Eq. (165)
we found $\left(\frac{dy_{2}}{dx}\right)_{x=1_{-}}=\frac{w}{\sqrt{w^{2}-4}}-1$
so that from Eq. (179) we have
$\left(\frac{du_{2}}{dx}\right)_{x=1_{+}}=\frac{w}{\sqrt{w^{2}-4}}-1$. Thus,
$\left(\frac{du_{2}}{dx}\right)_{x=1_{+}}\to\infty$ for $w\to 2$, and
$\left(\frac{du_{2}}{dx}\right)_{x=1_{+}}\to 0$ for $w\to\infty$. This was
stated in the main text.

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