EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-283 LHCb-PAPER-2016-047

Search for massive long-lived particles decaying semileptonically in the LHCb
detector

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

A search is presented for massive long-lived particles decaying into a muon
and two quarks. The dataset consists of proton-proton interactions at centre-
of-mass energies of 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$, corresponding to
integrated luminosities of 1 and 2$\mbox{\,fb}^{-1}$, respectively. The
analysis is performed assuming a set of production mechanisms with simple
topologies, including the production of a Higgs-like particle decaying into
two long-lived particles. The mass range from 20 to
80${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and lifetimes from 5 to
100${\mathrm{\,ps}}$ are explored. Results are also interpreted in terms of
neutralino production in different R-Parity violating supersymmetric models,
with masses in the 23–198 GeV/$c^{2}$ range. No excess above the background
expectation is observed and upper limits are set on the production cross-
section for various points in the parameter space of theoretical models.

Published on Eur. Phys. J. C (2017) 77:224

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

##  1 Introduction

Supersymmetry (SUSY) is one of the most popular extensions of the Standard
Model, which solves the hierarchy problem, can unify the gauge couplings and
could provide dark matter candidates. The minimal supersymmetric standard
model (MSSM) is the simplest phenomenologically viable realization of SUSY [1,
2]. The present study focuses on a subset of models featuring massive long-
lived particles (LLP) with a measurable flight distance [3, 4]. LLP searches
have been performed by Tevatron and LHC experiments [5, 6, 7, 8, 9, 10, 11],
often using the Hidden Valley framework [4] as a benchmark model (see also the
study of Ref. [12]). The LHCb detector probes the forward rapidity region
which is only partially covered by the other LHC experiments, and triggers on
particles with low transverse momenta, which allows the experiment to explore
relatively small LLP masses.

In this paper a search for massive long-lived particles is presented, using
proton-proton collision data collected by the LHCb detector at $\sqrt{s}=7$
and 8$\mathrm{\,Te\kern-1.00006ptV}$, corresponding to integrated luminosities
of 1 and 2 $\mbox{\,fb}^{-1}$, respectively. The event topology considered in
this study is a displaced vertex with several tracks including a high
$p_{\mathrm{T}}$ muon. This topology is found in the context of the minimal
super-gravity (mSUGRA) realisation of the MSSM, with R-parity violation [13],
in which the neutralino can decay into a muon and two jets. Neutralinos can be
produced by a variety of processes. In this paper four simple production
mechanisms with representative topologies and kinematics are considered, with
the assumed LLP mass in the range
20–80${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The LLP lifetime range
considered is 5–100${\mathrm{\,ps}}$, i.e. larger than the typical $b$-hadron
lifetime. It corresponds to an average flight distance of up to
30$\mathrm{\,cm}$, well inside the LHCb vertex detector. One of the production
mechanisms considered in detail is the decay into two LLPs of a Higgs-like
particle with an assumed mass between 50 and
130${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, i.e. in a range which includes
the mass of the scalar boson discovered by the ATLAS and CMS experiments [14,
15]. In addition, inclusive analyses are performed assuming the full set of
neutralino production mechanisms available in Pythia 6[16]. In this case the
LLP mass explored is in the range
23–198${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, inspired by Ref. [12], and
different combinations of gluino and squark masses are studied.

##  2 Detector description

The LHCb detector [17, 18] 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 (VELO), 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${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary vertex (PV), the impact parameter $d_{\rm IP}$, is measured with a
resolution of $(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam axis,
in ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. 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 [19], which consists of a
hardware stage based on information from the calorimeter and muon systems,
followed by a software stage which runs a simplified version of the offline
event reconstruction.

##  3 Event generation and detector simulation

Several sets of simulated events are used to design and optimize the signal
selection and to estimate the detection efficiency. Proton-proton collisions
are generated in Pythia 6 with a specific LHCb configuration [20], and with
parton density functions taken from CTEQ6L [21]. The LLP signal in this
framework is represented by the lightest neutralino $\tilde{\chi}^{0}_{1}$,
with mass $m_{\rm{LLP}}$ and lifetime $\tau_{\rm{LLP}}$. It is allowed to
decay into two quarks and a muon. Decays to all quark pairs are assumed to
have identical branching fractions except for those involving a top quark,
which are neglected.

Two separate detector simulations are used to produce signal models: a full
simulation, where the interaction of the generated particles with the detector
is based on Geant4 [22, *Agostinelli:2002hh], and a fast simulation. In
Geant4, the detector and its response are implemented as described in Ref.
[24]. In the fast simulation, which is used to cover a broader parameter space
of the theoretical models, the charged particles falling into the geometrical
acceptance of the detector are processed by the vertex reconstruction
algorithm. The simulation accounts for the effects of the material veto
described in the next section. The program also provides parameterised
particle momenta resolutions, but it is found that these resolutions have no
significant impact on the LLP mass reconstruction, nor on the signal detection
efficiency. The fast simulation is validated by comparison with the full
simulation. The distributions for mass, momentum and transverse momentum of
the reconstructed LLP and for the reconstructed decay vertex position are in
excellent agreement, as well as the muon momentum and its impact parameter to
the PV. The detection efficiencies predicted by the full and the fast
simulation differ by less than 5%.

$P\\!A$

$P\\!B$

$P\\!\,C$

$P\\!D$

Figure 1:  Four topologies considered as representative LLP production
mechanisms: $P\\!A$ non-resonant direct double LLP production, $P\\!B$ single
LLP production, $P\\!\,C$ double LLP production from the decay of a Higgs-like
boson, $P\\!D$ double LLP indirect production via squarks.

Two LLP production scenarios are considered. In the first, the signal samples
are generated assuming the full set of neutralino production processes
available in Pythia. In particular, nine models are fully simulated with the
parameters given in the Appendix, Table 4. Other points in the parameter space
of the theoretical models are studied with the fast simulation, covering the
$m_{\rm{LLP}}$ range 23–198${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. These
models are referred to as “LV” (for lepton number violation) followed by the
LLP mass in ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and lifetime (e.g. LV98
10${\mathrm{\,ps}}$). For the second scenario, the four production mechanisms
depicted in Fig. 1, labelled $P\\!A$, $P\\!B$, $P\\!\,C$, and $P\\!D$, are
selected and studied independently with the fast simulation. The LLP,
represented by the neutralino, subsequently decays into two quarks and a muon.
The processes $P\\!A$, $P\\!\,C$, and $P\\!D$ have two LLPs in the final
state. In processes $P\\!\,C$ and $P\\!D$ two LLPs are produced by the decay
of a Higgs-like particle of mass $m_{\rm h^{0}}$, and by the decay of squarks
of mass $m_{\tilde{\rm q}}$, respectively. In process $P\\!B$ a single LLP is
produced recoiling against an object labelled as a “gluino”, of mass
$m_{\rm``\tilde{g}"}$. In order to control the kinematic conditions, the
particles generated in these processes are constrained to be on-shell and the
“gluino” of option $P\\!B$ is stable. Since LHCb is most sensitive to
relatively low LLP masses, only $m_{\rm{LLP}}$ values below
80${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are considered.

The background from direct production of heavy quarks, as well as from ${W}$
and ${Z}$ boson decays, is studied using the full simulation. A sample of
$9\times 10^{6}$ inclusive ${c}{\overline{c}}$ events with at least two $c$
hadrons in $1.5<\eta<5.0$, and another sample of about $5\times 10^{5}$
${t}{\overline{t}}$ events with at least one muon in $1.5<\eta<5.0$ and
$\mbox{$p_{\mathrm{T}}$}>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ were
produced. Several million simulated events are available with production of
${W}$ and ${Z}$ bosons. The most relevant background in this analysis is from
${b}{\overline{b}}$ events. The available simulated inclusive
${b}{\overline{b}}$ events are not numerous enough to cover the
high-$p_{\mathrm{T}}$ muon kinematic region required in this analysis. To
enhance the ${b}{\overline{b}}$ background statistics, a dedicated sample of
$2.14\times 10^{5}$ simulated events has been produced with a minimum parton
$\hat{p}_{\rm T}$ of 20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and requiring a
muon with $\mbox{$p_{\mathrm{T}}$}>12{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in
$1.5<\eta<5.0$. As a consequence of limitations in the available computing
power, only ${b}{\overline{b}}$ events with
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ have been fully simulated. Despite
the considerable increase of generation efficiency, all the simulated
${b}{\overline{b}}$ events are rejected by the multivariate analysis presented
in the next section. Therefore a data-driven approach is employed for the
final background estimation.

##  4 Event selection

Signal events are selected by requiring a displaced high-multiplicity vertex
with one associated isolated high-$p_{\mathrm{T}}$ muon, since, due to the
larger particle mass, muons from LLP decays are expected to have larger
transverse momenta and to be more isolated than muons from hadron decays.

The events from $pp$ collisions are selected online by a trigger requiring
muons with $\mbox{$p_{\mathrm{T}}$}>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Primary vertices and displaced vertices are reconstructed offline from charged
particle tracks [25] with a minimum reconstructed $p_{\mathrm{T}}$ of
100${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Genuine PVs are identified by a
small radial distance from the beam axis, $R_{\rm xy}<0.3$ mm. The offline
analysis requires that the triggering muon has an impact parameter to all PVs
of $d_{\rm IP}>0.25\mathrm{\,mm}$ and
$\mbox{$p_{\mathrm{T}}$}>12$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. To
suppress the background due to kaons or pions punching through the
calorimeters and being misidentified as muons, the corresponding energy
deposit in the calorimeters must be less than 4% of the muon energy. To
preserve enough background events in the signal-free region for the signal
determination algorithm described in Sect. 5, no isolation requirement is
applied at this stage. Secondary vertices are selected by requiring $R_{\rm
xy}>0.55$ mm, at least four tracks in the forward direction (i.e. in the
direction of the spectrometer) including the muon and no tracks in the
backward direction. The total invariant mass of the tracks coming from a
selected vertex must be larger than
4.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Particles interacting with the
detector material are an important source of background. A geometric veto is
used to reject events with vertices in regions occupied by detector material
[26].

The number of data events selected is 18 925 (53 331) in the $\rm 7\,TeV$
($\rm 8\,TeV$) datasets. Less than 1% of the events have more than one
candidate vertex, in which case the candidate with the
highest-$p_{\mathrm{T}}$ muon is chosen. According to the simulation, the
background is largely dominated by ${b}{\overline{b}}$ events, while the
contribution from the decays of ${W}$ and ${Z}$ bosons is of the order of 10
events. All simulated ${c}{\overline{c}}$ and ${t}{\overline{t}}$ events are
rejected. The ${b}{\overline{b}}$ cross-section value measured by LHCb,
$288\pm 4\pm 48$ $\mu$b [27, 28], predicts $(15\pm 3)\times 10^{3}$ events for
the $\rm 7\,TeV$ dataset, after selection. The value for the $\rm 8\,TeV$
dataset is $(52\pm 10)\times 10^{3}$. The extrapolation of the cross-section
from $\rm 7\,TeV$ to $\rm 8\,TeV$ is obtained from POWHEG [29, *powheg1,
*powheg2], while Pythia is used to obtain the detection efficiency. The
candidate yields for the two datasets are consistent with a dominant
${b}{\overline{b}}$ composition of the background. This is confirmed by the
study of the shapes of the distributions of the relevant observables. Figure 2
compares the distributions for the $\rm 7\,TeV$ dataset and for the 135
simulated ${b}{\overline{b}}$ events surviving the selection. For
illustration, the shapes of simulated LV38 10${\mathrm{\,ps}}$ signal events
are superimposed on all the distributions, as well as the expected shape for
LV38 50${\mathrm{\,ps}}$ on the $R_{\rm xy}$ distribution. The muon isolation
variable is defined as the sum of the energy of tracks surrounding the muon
direction, including the muon itself, in a cone of radius $R_{\eta\phi}=0.3$
in the pseudorapidity-azimuthal angle $(\eta,\phi)$ space, divided by the
energy of the muon track. The corresponding distribution is shown in Fig. 2
b). A muon isolation value of unity denotes a fully isolated muon. As
expected, the muon from the signal is found to be more isolated than the
hadronic background. Figure 2 e) presents the radial distribution of the
displaced vertices; the drop in the number of candidates with a vertex above
$R_{\rm xy}\sim 5\mathrm{\,mm}$ is due to the material veto. From simulation,
the veto introduces a loss of efficiency of 13% (42%) for the detection of
LLPs with a $30{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ mass and a
10${\mathrm{\,ps}}$ (100${\mathrm{\,ps}}$) lifetime. The radial ($\sigma_{\rm
R}$) and longitudinal ($\sigma_{\rm z}$, parallel to the beam) uncertainties
provided by the LLP vertex fit are shown in Figs. 2 f) and g). Larger
uncertainties are expected from the vertex fits of candidates from
${b}{\overline{b}}$ events compared to signal LLPs. The former are more
boosted and produce more narrowly collimated tracks, while the relatively
heavier signal LLPs decay into more divergent tracks. This effect decreases
when $m_{\rm{LLP}}$ approaches the mass of $b$-quark hadrons.

Figure 2:  Distributions for the 7 TeV dataset (black histogram) compared to
simulated ${b}{\overline{b}}$ events (blue squares with error bars), showing
a) transverse momentum and b) isolation of the muon, c) number of tracks of
the displaced vertex, d) reconstructed mass, e) radial position of the vertex,
f) and g) vertex fit uncertainties in the radial and z directions. The fully
simulated signal distributions for LV38 10${\mathrm{\,ps}}$ are shown (red
dashed histograms), as well as LV38 50${\mathrm{\,ps}}$ (green dotted
histogram) in e). The distributions from simulation are normalised to the
number of data entries.

A multivariate analysis based on a multi-layer perceptron (MLP) [32, 33] is
used to further purify the data sample. The MLP input variables are the muon
$p_{\mathrm{T}}$ and impact parameter, the number of charged particle tracks
used to reconstruct the LLP, the vertex radial distance $R_{\rm xy}$ from the
beam line, and the uncertainties $\sigma_{\rm R}$ and $\sigma_{\rm z}$
provided by the LLP vertex fit. The muon isolation value and the reconstructed
mass of the long-lived particles are not used in the MLP classifier; the
discrimination power of these two variables is subsequently exploited for the
signal determination. The signal training and test samples are obtained from
simulated signal events selected under the same conditions as data. A data-
driven approach is used to provide the background training samples, based on
the hypothesis that the amount of signal in the data is small. For this, a
number of candidates equal to the number of candidates of the signal training
set, which is of the order of 1000, is randomly chosen in the data. The same
procedure provides the background test samples. The MLP training is performed
independently for each fully simulated model and dataset. The optimal MLP
requirement is subsequently determined by maximizing a figure of merit defined
by $\epsilon/\sqrt{N_{d}+1}$, where $\epsilon$ is the signal efficiency from
simulation for a given selection, and $N_{d}$ the corresponding number of
candidates found in the data.

The generalisation power of the MLP is assessed by verifying that the
distributions of the classifier output for the training sample and the test
sample agree. The uniformity over the dataset is controlled by the comparison
of the MLP responses for several subsets of the data.

The MLP classifier can be biased by the presence of signal in the data events
used as background training set. To quantify the potential bias, the MLP
training is performed adding a fraction of simulated signal events (up to 5%)
to the background set. This test, performed independently for all signal
models, demonstrates a negligible variation of the performances quantified by
the above figure of merit.

##  5 Determination of the signal yield

The signal yield is determined with an extended unbinned maximum likelihood
fit to the distribution of the reconstructed LLP mass, with the shape of the
signal component taken from the simulated models, plus the background
component. After the MLP filter, no simulated background survives; therefore a
data-driven method is adopted to determine the background template. The data
candidates are separated into a signal region with muon isolation below 1.4
and a background region with isolation value from 1.4 to 2. The signal region
contains more than 90% of the signal for all the models considered (see e.g.
Fig. 2 (b)). The reconstructed mass obtained from the background candidates is
used to constrain an empirical probability density function (PDF) consisting
of the sum of two negative slope exponential functions, for which the slope
values and amplitudes are free parameters in the fit. The signal PDF is taken
from the histogram of the mass distribution obtained from simulation. The fit
is performed simultaneously on events from the background region and from the
signal region. In the latter the numbers of signal and background events are
left free in the fit, while the slope values and the relative strength of the
two exponential functions are in common with the background region fit.
Examples of fit results are given in Fig. 3, obtained from the $\rm 8\,TeV$
dataset for two signal hypotheses, LV38 5${\mathrm{\,ps}}$ and LV98
10${\mathrm{\,ps}}$. The fitted signal yields, given in Table 1, for both
datasets are compatible with the background-only hypothesis.

Figure 3: Reconstructed mass of the LLP candidate from the 8 TeV dataset. The top plots correspond to events with candidates selected from the background region of the muon isolation variable. They are fitted with the sum of two exponential functions. In the bottom row the candidates from the signal region are fitted including a specific signal shape, added to the background component. Subfigures a) and c) correspond to the analysis which assumes the LV38 5${\mathrm{\,ps}}$ signal model, b) and d) are for LV98 10${\mathrm{\,ps}}$.  Table 1: Total signal detection efficiency $\epsilon$, including the geometrical acceptance, and numbers of fitted signal and background events, $N_{\rm s}$ and $N_{\rm b}$, for the different signal hypotheses. The last column gives the value of $\chi^{2}/\mathrm{ndf}$ from the fit. The signal models are from the full simulation. Uncertainties are explained in Sect. 6.  Dataset | Model |  $\epsilon$ % | $N_{\rm b}$ | $N_{\rm s}$ | $\chi^{2}/\mathrm{ndf}$
---|---|---|---|---|---
$\rm 7\,TeV$ |  LV 38 5${\mathrm{\,ps}}$ | $0.52\pm 0.03$ | $140.2\pm 15.5$ | $\phantom{-}3.8\pm 10.0$ | 0.64
|  LV 38 10${\mathrm{\,ps}}$ | $0.57\pm 0.03$ | $115.2\pm 13.3$ | $\phantom{-}4.8\pm\phantom{0}8.2$ | 1.71
|  LV 38 50${\mathrm{\,ps}}$ | $0.43\pm 0.02$ | $112.9\pm 13.3$ | $\phantom{-}9.0\pm\phantom{0}8.6$ | 1.50
|  LV 98 5${\mathrm{\,ps}}$ | $0.58\pm 0.03$ | $\phantom{0}97.3\pm 10.3$ | $-3.3\pm\phantom{0}2.4$ | 0.88
|  LV 98 10${\mathrm{\,ps}}$ | $0.72\pm 0.04$ | $\phantom{0}62.6\pm\phantom{0}8.7$ | $-5.6\pm\phantom{0}2.8$ | 1.06
|  LV 98 50${\mathrm{\,ps}}$ | $0.56\pm 0.03$ | $\phantom{0}99.9\pm 11.2$ | $-3.9\pm\phantom{0}4.5$ | 0.33
|  LV198 5${\mathrm{\,ps}}$ | $0.60\pm 0.04$ | $143.8\pm 12.5$ | $-6.9\pm\phantom{0}2.2$ | 1.42
|  LV198 10${\mathrm{\,ps}}$ | $0.76\pm 0.04$ | $158.1\pm 13.1$ | $-6.1\pm\phantom{0}2.7$ | 1.63
|  LV198 50${\mathrm{\,ps}}$ | $0.66\pm 0.04$ | $118.8\pm 11.3$ | $-0.9\pm\phantom{0}2.8$ | 0.89
$\rm 8\,TeV$ |  LV 38 5${\mathrm{\,ps}}$ | $0.54\pm 0.04$ | $120.3\pm 15.6$ | $\phantom{-}2.9\pm\phantom{0}9.5$ | 0.74
|  LV 38 10${\mathrm{\,ps}}$ | $0.66\pm 0.04$ | $203.7\pm 19.9$ | $-1.6\pm 13.3$ | 0.81
|  LV 38 50${\mathrm{\,ps}}$ | $0.43\pm 0.02$ | $123.3\pm 15.6$ | $\phantom{-}3.7\pm 11.0$ | 0.99
|  LV 98 5${\mathrm{\,ps}}$ | $0.77\pm 0.05$ | $121.0\pm 11.2$ | $\phantom{-}1.0\pm\phantom{0}2.2$ | 1.26
|  LV 98 10${\mathrm{\,ps}}$ | $0.96\pm 0.05$ | $123.7\pm 12.0$ | $\phantom{-}2.4\pm\phantom{0}3.4$ | 0.74
|  LV 98 50${\mathrm{\,ps}}$ | $0.69\pm 0.04$ | $103.8\pm 10.5$ | $\phantom{-}2.2\pm\phantom{0}2.8$ | 0.94
|  LV198 5${\mathrm{\,ps}}$ | $0.79\pm 0.06$ | $196.3\pm 14.2$ | $-2.3\pm\phantom{0}2.0$ | 1.94
|  LV198 10${\mathrm{\,ps}}$ | $1.06\pm 0.07$ | $258.7\pm 16.2$ | $\phantom{-}2.3\pm\phantom{0}2.3$ | 1.53
|  LV198 50${\mathrm{\,ps}}$ | $0.69\pm 0.04$ | $113.7\pm 10.8$ | $\phantom{-}1.3\pm\phantom{0}2.1$ | 1.73

The validity of using events with isolation above 1.4 to model the background
has been checked by comparing the relevant distributions from events in the
background and in the signal regions, including the muon $p_{\mathrm{T}}$ and
impact parameter distributions, as well as the number of tracks, invariant
mass, vertex $R_{\rm xy}$ and vertex uncertainties of the LLP candidate. This
test is performed with the nominal MLP selection, and also with loosened
requirements that result in a threefold increase in the number of background
candidates. In both cases all distributions agree within statistical
uncertainties, with the $\chi^{2}/\mathrm{ndf}$ of the comparison in the range
0.6-1.5.

The sensitivity of the procedure is studied by adding a small number of signal
events to the data according to a given signal model. The fitted yields are
consistent with the numbers of added events on average, and the pull
distributions are close to Gaussian functions with mean values between $-0.1$
and $0.1$ and standard deviations in the range from 0.9 to 1.2.

As a final check a two-dimensional sideband subtraction method (“ABCD method”
[34]) has been considered. The LLP reconstructed mass and the muon isolation
are used to separate the candidates in four regions. The results of this check
are also consistent with zero signal for the two datasets.

Both the LLP mass fit and the ABCD methods are tested with $W$ and
${Z}/\gamma$ leptonic decays. Isolated high-$p_{\mathrm{T}}$ muons are
produced in such processes with kinematic properties similar to the signal. By
removing the minimum $R_{\rm xy}$ requirement the candidates can be formed by
collecting tracks from the primary vertex. As before, the background is taken
from a region of muon isolation above 1.4, which contains a negligible amount
of signal. For both datasets the number of events obtained from this study is
compatible with the cross-sections measured by LHCb [35, 36, 37].

##  6 Detection efficiency and systematic uncertainties

The total signal detection efficiency, estimated from fully simulated events,
is shown in Table 1. It includes the geometrical acceptance, which for the
detection of one $\tilde{\chi}^{0}_{1}$ in LHCb is about 11% (12%) at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ (8$\mathrm{\,Te\kern-1.00006ptV}$).
The efficiencies for the models where the fast simulation is used, including
processes $P\\!A$, $P\\!B$, $P\\!\,C$, and $P\\!D$, vary from about 0.1% to
about 2%. The efficiency increases with $m_{\rm{LLP}}$ because more particles
are produced in the decay of heavier LLPs. This effect is only partially
counteracted by the loss of particles outside the spectrometer acceptance,
which is especially likely when the LLP are produced from the decay of heavier
states, such as the Higgs-like particles of process $P\\!\,C$. Another
competing phenomenon is that the lower boost of heavier LLPs results in a
shorter average flight length, i.e. the requirement of a minimum $R_{\rm xy}$
disfavours heavy LLPs. The cut on $R_{\rm xy}$ is more efficient at selecting
LLPs with large lifetimes, but for lifetimes larger than $\sim
50{\mathrm{\,ps}}$ a considerable portion of the decays falls into the
material region and is vetoed. Finally, a drop of sensitivity is expected for
LLPs with a lifetime close to the $b$ hadron lifetimes, where the
contamination from ${b}{\overline{b}}$ events becomes important, especially
for low mass LLPs.

A breakdown of the relative systematic uncertainties for the analysis of the
$\rm 8\,TeV$ dataset is shown in Table 2. The table does not account for the
uncertainties associated with the fit procedure, which, as described below,
require a specific treatment. The uncertainties on the integrated luminosity
are $1.7\%$ for $\rm 7\,TeV$ dataset and $1.2\%$ for $\rm 8\,TeV$ data [38].
Several sources of systematic uncertainty coming from discrepancies between
data and simulation have been considered.

The muon detection efficiency, including trigger, tracking, and muon
identification efficiencies, is studied by a tag-and-probe technique applied
to muons from $J/\psi\rightarrow{\mu^{+}\mu^{-}}$ [39] and from
${Z}\rightarrow{\mu^{+}\mu^{-}}$ decays [40, 37, 36, 35]. The corresponding
systematic effects due to differences between data and simulation are
estimated to be between 2.1% and 4.5%, depending on the theoretical model
considered.

A comparison of the simulated and observed $p_{\mathrm{T}}$ distributions of
muons from ${Z}\rightarrow{\mu^{+}\mu^{-}}$ decays shows a maximum difference
of 3% in the momentum scale; this difference is propagated to the LLP analysis
by moving the muon $p_{\mathrm{T}}$ threshold by the same amount. A
corresponding systematic uncertainty of 1.5% is estimated for all models under
consideration.

The $d_{\rm IP}$ distribution shows a discrepancy between data and simulation
of about 5${\,\upmu\mathrm{m}}$ in the mean value for muons from ${Z}$ decays,
with a maximum deviation of about 20${\,\upmu\mathrm{m}}$ close to the muon
$p_{\mathrm{T}}$ threshold. By changing the minimum $d_{\rm IP}$ requirement
by this amount, the change in the detection efficiency is in the range
0.4–1.2%, depending on the model.

The vertex reconstruction efficiency is affected by the tracking efficiency
and has a complicated spatial structure due to the geometry of the VELO and
the material veto. In the material-free region, $R_{\rm xy}<4.5\mathrm{\,mm}$,
the efficiency to detect secondary vertices as a function of the flight
distance has been studied in detail, in particular in the context of the $b$
hadrons lifetime measurement [41]. The deviation of the efficiency in
simulation with respect to the data is below 1%. For $R_{\rm xy}$ from
4.5$\mathrm{\,mm}$ to about 12$\mathrm{\,mm}$ a study performed with inclusive
${b}{\overline{b}}$ events finds differences between data and simulation of
less than 5%. The corresponding systematic uncertainties are determined by
altering the efficiency in the simulation program as a function of the true
vertex position. A maximum of 1% uncertainty is obtained for all the signal
models. An alternative procedure to asses this uncertainty considers vertices
from $B^{0}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decays with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}}$ and ${{K}^{*0}}\rightarrow
K^{+}\pi^{-}$. The detection efficiency in data and simulation is found to
agree within 10%. This result, obtained from a four-particle final state, when
propagated to LLP decays with on average more than 10 charged final-state
particles for all modes, results in a discrepancy of at most 2% between the
LLP efficiencies in data and simulation, which is the adopted value for the
respective systematic uncertainty.

The uncertainty on the position of the beam line is less than
20${\,\upmu\mathrm{m}}$ [42]. It can affect the secondary vertex selection,
mainly via the requirement on $R_{\rm xy}$. By altering the PV position in
simulated signal events, the maximum effect on the LLP selection efficiency is
in the range 0.2–1%.

The imprecision of the models used for training the MLP propagates into a
systematic difference of the detection efficiency between data and simulation.
The bias on each input variable is determined by comparing simulated and
experimental distributions for muons and LLP candidates from $Z$ and $W$
events, and from ${b}{\overline{b}}$ events. The effect of the biases is
subsequently estimated by testing the trained classifier on altered simulated
signal events: each input variable is modified by a scale factor randomly
drawn from a Gaussian distribution of width equal to the corresponding bias.
The RMS variation of the signal efficiency distributions after the MLP range
from 1.5 to 3.6% depending on the signal model. These values are taken as
contributions to the systematic uncertainties.

The signal region is selected by the requirement of a muon isolation value
lower than 1.4. By a comparison of data and simulated muons from $Z$ decays,
the uncertainty on this variable is estimated to be $\pm 0.05$. This
uncertainty is propagated to a maximum 2.2% effect on the detection
efficiency.

Comparing the mass distributions of ${b}{\overline{b}}$ events selected with
relaxed cuts, a maximum mass scale discrepancy between data and simulated
events of 10% is estimated. The corresponding shift of the simulated signal
mass distribution results in a variation of the detection efficiency between
0.8 and 1.5%.

The statistical precision of the efficiency value determined from the
simulated events is in the range 1.7–2.5% for the different models.

The theoretical uncertainties are dominated by the uncertainty of the partonic
luminosity. Their contribution to the detection efficiency uncertainty is
estimated following the procedure explained in Ref. [43] and vary from 3% up
to a maximum of 7%, which is found for the gluon-gluon fusion process
$P\\!\,C$.

For the analysis based on the fast simulation, a 5% uncertainty is added to
account for the difference between the fast and the full simulation, as
explained in Sect. 3.

Table 2: Summary of the contributions to the relative systematic
uncertainties, corresponding to the 8 TeV dataset, (the sub-total for the 7
TeV dataset is also given). The indicated ranges cover the fully simulated LV
models. The detection efficiency is affected by the parton luminosity model
and depends upon the production process, with a maximum uncertainty of 7% for
the gluon-gluon fusion process $P\\!\,C$. For the fast simulation based
analysis there is an additional contribution of 5%. The systematic effects
associated with the signal and background models used in the LLP mass fit are
not shown in the table.

Source | Contribution (%)
---|---
Integrated luminosity | 1.2
Muon detection | $2.1-4.5$
Muon $p_{\mathrm{T}}$ scale | $1.5$
Muon $d_{\rm IP}$ uncertainty | $0.4-1.2$
Vertex reconstruction | 2.0
Beam line uncertainty | $0.2-1.0$
MLP training models | $1.5-3.6$
Muon isolation | $2.2$
LLP mass scale | $0.8-1.5$
Models statistics | $1.7-2.5$
Sub-total $\rm 8\,TeV$ dataset | $4.9-6.5$
(Sub-total $\rm 7\,TeV$ dataset |  $4.9-6.1$)
Parton luminosity | $3-7$
Analysis with fast simulation | $5$

The choice of the background and signal templates can affect the results of
the LLP mass fit. The uncertainty due to the signal model accounts for the
mass scale, the mass resolution and the finite number of events available to
construct the model. Pseudoexperiments in which 10 signal events are added to
the data are analysed with a modified signal template, and the resulting
number of fitted candidates is compared to the result from the nominal fit
model. Assuming as before a 10% uncertainty on the signal mass scale, a
maximum absolute variation of 0.6 fitted signal candidates is obtained. No
significant effects are obtained by modifying the signal mass resolution with
an additional smearing. Changing the statistical precision by reducing the
initial number $N$ of signal events used to build the histogram PDF by
$2\sqrt{N}$ has no significant effect either.

The uncertainty induced by the choice of the background model is obtained by
reweighting the candidates from the background region in such a way that the
distribution of the number of tracks included in the LLP vertex fit exactly
matches the distribution in the signal region. This test is motivated by the
fact that the number of tracks has a significant correlation with the measured
mass. The fits of the mass distribution of pseudoexperiments give absolute
variations in the numbers of fitted signal events in the range 0.1–1.6, the
largest value at low LLP mass. Reweighting the candidates in such a way as to
match the $p_{\mathrm{T}}$ distributions gives variations which are less than
0.5 events for all models. Moving the isolation threshold by $\pm 0.1$ leads
to variations of the order of 0.01 events. In conclusion, the variation on the
number of fitted candidates associated to the choice of the PDF models is in
the range of 1–2 events. The calculation of the cross-section upper limits
takes into account this uncertainty as an additional nuisance parameter on the
fit procedure.

##  7 Results

The LLP candidates collected at $\sqrt{s}=7$ and
8$\mathrm{\,Te\kern-1.00006ptV}$ are analysed independently. The fast
simulation is used to extend the MSSM/mSUGRA theoretical parameter space of
the LV models, and for the analysis of processes $P\\!A$, $P\\!B$, $P\\!\,C$,
and $P\\!D$. The results obtained are found to be compatible with the absence
of signal for all signal model hypotheses considered. The 95% confidence level
(CL) upper limit on the production cross-sections times branching fraction is
computed for each model using the CLs approach [44]. The numerical results for
the fully simulated LV models are given in Table 3 111The numerical results
for all the other models are available as supplementary material. A graphical
representation of selected results is given in Figs. 4 to 6.

The MSSM/mSUGRA LV models are explored by changing the common squark mass and
the gluino mass. Figure 4 gives examples of the cross-section times branching
fraction upper limits as a function of $m_{\rm{LLP}}$ for such models for two
values of $\tau_{\rm{LLP}}$, and two values of the squark mass. The gluino
mass is set to 2000${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Varying the
gluino mass from 1500 to 2500${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ has
almost no effect on the results. The decrease of sensitivity for decreasing
$m_{\rm{LLP}}$ is explained by the above-mentioned effects on the detection
efficiency.

A representation of selected results from the processes $P\\!A$, $P\\!B$,
$P\\!\,C$, and $P\\!D$ is given in Fig. 5. The single LLP production of
$P\\!B$ has a lower detection probability compared to the double LLP
production case, $P\\!A$, which explains the reduced sensitivity. The $P\\!B$
plots correspond to
$m_{\rm``\tilde{g}"}=100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Varying
$m_{\rm``\tilde{g}"}$ from 100 to
1000${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ decreases the detection
efficiency by a factor of two, while an increase by a factor of two is
obtained reducing $m_{\rm``\tilde{g}"}$ to
20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The results for process
$P\\!\,C$ are given as a function of the Higgs-like boson mass, for three
values of $m_{\rm{LLP}}$. Again the sensitivity of the analysis drops with
decreasing $m_{\rm{LLP}}$. The results shown for $P\\!D$ are for
$m_{\tilde{\rm q}}=60{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, which limits
the maximum $m_{\rm{LLP}}$ value. In process $P\\!D$ some of scattering energy
is absorbed by an additional jet during the LLP production, reducing the
detection efficiency by a factor of two with respect to $P\\!A$. Finally, Fig.
6 gives the cross-section upper limits times branching fraction as a function
of $m_{\rm{LLP}}$, for the process $P\\!\,C$ with a mass of
125${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for the Higgs-like boson and LLP
lifetime from 5 to 100${\mathrm{\,ps}}$. These results can be compared to the
prediction of the Standard Model Higgs production cross-section of about
21$\mathrm{\,pb}$ at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ [45].

Table 3:  Upper limits (95% CL) on the production cross-section times
branching fraction (pb) for the $\rm 7\,TeV$ and $\rm 8\,TeV$ datasets, based
on the fully simulated LV signal samples.

|  $\rm 7\,TeV$ dataset |  |  $\rm 8\,TeV$ dataset
---|---|---|---
Model | Expected | Observed |  | Expected | Observed
LV 38 5${\mathrm{\,ps}}$ | 4.03${}^{+1.79}_{-1.20}$ | 4.73 |  | 2.04${}^{+0.89}_{-0.60}$ | 2.32
LV 38 10${\mathrm{\,ps}}$ | 2.95${}^{+1.36}_{-0.89}$ | 3.76 |  | 2.24${}^{+0.95}_{-0.65}$ | 2.13
LV 38 50${\mathrm{\,ps}}$ | 4.08${}^{+1.89}_{-1.24}$ | 6.15 |  | 2.86${}^{+1.23}_{-0.83}$ | 3.10
LV 98 5${\mathrm{\,ps}}$ | 1.78${}^{+0.97}_{-0.60}$ | 1.21 |  | 0.62${}^{+0.36}_{-0.22}$ | 0.57
LV 98 10${\mathrm{\,ps}}$ | 1.52${}^{+0.78}_{-0.49}$ | 0.94 |  | 0.52${}^{+0.27}_{-0.17}$ | 0.53
LV 98 50${\mathrm{\,ps}}$ | 2.21${}^{+1.10}_{-0.70}$ | 1.83 |  | 0.70${}^{+0.41}_{-0.25}$ | 0.77
LV198 5${\mathrm{\,ps}}$ | 1.50${}^{+0.86}_{-0.52}$ | 0.95 |  | 0.59${}^{+0.34}_{-0.21}$ | 0.40
LV198 10${\mathrm{\,ps}}$ | 1.18${}^{+0.68}_{-0.41}$ | 0.85 |  | 0.27${}^{+0.20}_{-0.11}$ | 0.42
LV198 50${\mathrm{\,ps}}$ | 0.92${}^{+0.67}_{-0.38}$ | 1.07 |  | 0.52${}^{+0.35}_{-0.21}$ | 0.58

Figure 4:  Expected (open dots with 1$\sigma$ and 2$\sigma$ bands) and
observed (full dots) cross-section times branching fraction upper limits at
95% confidence level, as a function of the LLP mass from the 8 TeV dataset.
The theoretical models assume the full set of SUSY production processes
available in Pythia 6 with default parameter settings, unless otherwise
specified. The gluino mass is 2000${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$.

Figure 5:  Expected (open dots and 1$\sigma$ and 2$\sigma$ bands) and observed
(full dots) cross-section times branching fraction upper limits (95% CL) for
the processes indicated in the bottom left corner of each plot,
$\tau_{\rm{LLP}}$ is always 10${\mathrm{\,ps}}$. The results correspond to the
8 TeV dataset. a) upper limits as a function of the LLP mass for process
$P\\!A$; b) as a function of the LLP mass for process $P\\!B$, with
$m_{\rm``\tilde{g}"}=100$${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$; c) as a
function of $m_{\rm h^{0}}$ for process $P\\!\,C$ for $m_{\rm{LLP}}$ of 20,
40, and 60${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$, from top to bottom (the
single point at 130${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ with
$m_{\rm{LLP}}=60{\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ has been shifted to
the right for visualisation); d) upper limits as a function of the LLP mass
for process $P\\!D$ with $m_{\tilde{\rm
q}}=60$${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$.

Figure 6:  Expected (open dots with 1$\sigma$ and 2$\sigma$ bands) and
observed (full dots) cross-section times branching fraction upper limits (95%
CL) for the processes $P\\!\,C$ as a function of the LLP mass; the LLP
lifetime $\tau_{\rm{LLP}}$ is indicated in each plot, $m_{\rm
h^{0}}=125{\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}.$ The results correspond to
the 8 TeV dataset.

##  8 Conclusion

Long-lived massive particles decaying into a muon and two quarks have been
searched for using proton-proton collision data collected by LHCb at
$\sqrt{s}=7$ and 8$\mathrm{\,Te\kern-1.00006ptV}$, corresponding to integrated
luminosities of 1 and 2 $\mbox{\,fb}^{-1}$, respectively. The background is
dominated by ${b}{\overline{b}}$ events and is reduced by tight selection
requirements, including a dedicated multivariate classifier. The number of
candidates is determined by a fit to the LLP reconstructed mass with a signal
shape inferred from the theoretical models.

LHCb can study the forward region $2<\eta<5$, and its low trigger
$p_{\mathrm{T}}$ threshold allows the experiment to explore relatively small
LLP masses. The analysis has been performed assuming four LLP production
mechanisms with the topologies shown in Fig. 1, covering LLP lifetimes from 5
ps up to 100${\mathrm{\,ps}}$ and masses in the range 20–80
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. One of the processes proceeds via
the decay of a Higgs-like particle into two LLPs: the mass of the Higgs-like
particle is varied between 50${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
130${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, comprising the mass of the
scalar boson discovered by the ATLAS and CMS experiments. In addition, the
full set of neutralino production mechanisms available in Pythia in the
context of MSSM/mSUGRA has been considered, with an LLP mass range 23–198
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The results for all theoretical
models considered are compatible with the background-only hypothesis. Upper
limits at 95% CL are set on the cross-section times branching fractions.

## 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).

Appendix

## Parameters of the fully simulated signal models

The parameters used to generate nine fully simulated signal samples in the
context of MSSM/mSUGRA are given in Table 4. Other MSSM parameters remain at
their default Pythia values. The lightest neutralino, $\tilde{\chi}^{0}_{1}$,
decays via the lepton number violating mode LQD (for the definition see [3,
13]). As an approximation, equal branching fractions are assumed for all QD
pairs, except for the pairs with a top quark, which are excluded.

Two sets of events have been produced with $\sqrt{s}$ =7 and
8$\mathrm{\,Te\kern-1.00006ptV}$. Only events with one muon and one
$\tilde{\chi}^{0}_{1}$ in the LHCb acceptance are processed in Geant4,
corresponding to about 11% of the $\sqrt{s}$ =7$\mathrm{\,Te\kern-1.00006ptV}$
generated events, 12% at 8$\mathrm{\,Te\kern-1.00006ptV}$.

Table 4:  Parameters for the generation of the nine fully simulated signal
models. The LLP is the lightest neutralino, $\tilde{\chi}^{0}_{1}$ with
$m_{\tilde{\chi}^{0}_{1}}=m_{\rm{LLP}}$; $M_{1}$ and $M_{2}$ are the Pythia
parameters RMSS(1) and RMSS(2), $m_{\tilde{\rm g}}$ is RMSS(3), $\mu$ is
RMSS(4), $\tan{\beta}$ RMSS(5) and $m_{\tilde{\rm q}}$ is RMSS(8-12). Samples
with lifetime of 5, 10 and 50${\mathrm{\,ps}}$ have been produced for each
mass.

Model | $M_{1}$ | $M_{2}$ | $m_{\tilde{\rm g}}$ | $\tan{\beta}$ | $\mu$ | $m_{\tilde{\rm q}}$ | $m_{\tilde{\chi}^{0}_{1}}$
---|---|---|---|---|---|---|---
| $[{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}]$ | $[{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}]$ | [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ] |  |  | [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ] | [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ]
LV38 5/10/50ps | 40 | 2000 | 2000 | 2.0 | 1200 | 1300 | 38
LV98 5/10/50ps | 100 | 2000 | 2000 | 2.0 | 1200 | 1300 | 98
LV198 5/10/50ps | 200 | 2000 | 2000 | 2.0 | 1200 | 1300 | 198

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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,
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G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B. Appleby56, F.
Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M. Artuso61, E.
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Zavertyaev11,c, L. Zhang3, Y. Zhang7, Y. Zhang63, A. Zhelezov12, Y. Zheng63,
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
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 38
70Van 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


# On the speed of convergence of Piterbarg constants

Krzysztof Bisewski  Krzysztof Bisewski, Department of Actuarial Science,
University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland
<EMAIL_ADDRESS>and  Grigori Jasnovidov  Grigori Jasnovidov,
Laboratory of Statistical Methods, St. Petersburg Department of Steklov
Mathematical Institute of Russian Academy of Sciences, Department of Actuarial
Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland
<EMAIL_ADDRESS>
Abstract: In this paper we derive an upper bound for the difference between
the continuous and discrete Piterbarg constants. Our result allows us to
approximate the classical Piterbarg constants by their discrete counterparts
using Monte Carlo simulations with an explicit error rate.

Key Words: fractional Brownian motion; Piterbarg constants; discretization
error; Monte Carlo simulation;

AMS Classification: 60G15; 60G70; 65C05

##  1\. Introduction

Let $B_{\alpha}(t),t\in\mathbb{R}$ be a fractional Brownian motion (later on
fBM), i.e., a Gaussian process with zero expectation and covariance function
given by

$\displaystyle\operatorname*{\mathbb{C}ov}(B_{\alpha}(t),B_{\alpha}(s))=\frac{|t|^{\alpha}+|s|^{\alpha}-|t-s|^{\alpha}}{2},\
\ \ \ \ t,s\in\mathbb{R},\ \alpha\in(0,2).$

One of the constants that typically appear in the asymptotics of the ruin
probabilities for Gaussian processes are the Piterbarg constants defined for
$d>0$ and $\alpha\in(0,2)$ by

$\displaystyle\mathcal{P}_{\alpha}(d,\mathcal{K})=\mathbb{E}\left\\{\sup\limits_{t\in\mathbb{R}\cap\mathcal{K}}e^{\sqrt{2}B_{\alpha}(t)-(1+d)|t|^{\alpha}}\right\\},$

where $\mathcal{K}$ can be both $[0,\infty)$ and $(-\infty,\infty)$. We refer
to Theorem 10.1 in [16] for finiteness and positivity of
$\mathcal{P}_{\alpha}(d,\mathcal{K})$; some other contributions dealing with
Piterbarg constants are [1, 15, 7, 13, 9]. We notice, that the Piterbarg
constants appear in Gaussian queueing theory results, see [11, 8]. As for
well-known Pickands constants (see, e.g., [16]), the exact value of the
classical Piterbarg constant is known only for special case $\alpha=1$, namely
$\mathcal{P}_{1}(d,(-\infty,\infty))=1+\frac{2}{d}-\frac{1}{2d+1}$ and
$\mathcal{P}_{1}(d,[0,\infty))=1+\frac{1}{d}$, see [1] and [14], respectively.
In other cases naturally arises the question of an approximation of the
classical Piterbarg constants.

Since it seems difficult to simulate fBM on a continuous-time scale, one can
approximate the Piterbarg constants by their discrete analogous defined for
$\delta>0$ by

$\displaystyle\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K})=\mathbb{E}\left\\{\sup\limits_{t\in
G(\delta)\cap\mathcal{K}}e^{\sqrt{2}B_{\alpha}(t)-(1+d)|t|^{\alpha}}\right\\},$

where $G(\delta)=\delta\mathbb{Z}$ for $\delta>0$. Set in the following
$\mathcal{P}_{\alpha}^{0}(d,\mathcal{K})=\mathcal{P}_{\alpha}(d,\mathcal{K})$
and $G(0)=\mathbb{R}$.

The question of speed of convergence of the discrete Piterbarg constants to
continuous ones is related to the estimation of
$\sup_{t\in[0,1]}B_{\alpha}(t)-\sup_{t\in[0,1]_{\delta}}B_{\alpha}(t)$ as
$\delta\to 0$. We refer to [5, 4] for the interesting analysis of the
expression above. For Brownian motion case (later on BM), i.e., when
$\alpha=1$, we refer to [10] for the survey of the known results for the
current moment.

##  2\. Main Results

Here we present the results needed for approximation of
$\mathcal{P}_{\alpha}(d,\mathcal{K})$ by simulation methods. The theorem below
derives an upper bound for the difference between the continuous and discrete
Piterbarg constants.

######  Theorem 2.1.

For any $d>0$ with some constant $\mathcal{C}>0$ that does not depend on
$\delta$ it holds, that as $\delta\to 0$

$\displaystyle\mathcal{P}_{\alpha}(d,\mathcal{K})-\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K})\leq\mathcal{C}\delta^{\alpha/2}(-\ln\delta)^{1/2}.$

Since it is impossible to simulate a fBM on the infinite-time horizon we deal
with the truncated version of the Piterbarg constants. Namely, for $d>0$ and
$\delta,T\geq 0$ a truncated Piterbarg constant is defined by

$\displaystyle\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K};T)=\mathbb{E}\left\\{\sup\limits_{t\in\mathcal{K}\cap[-T,T]_{\delta}}e^{\sqrt{2}B_{\alpha}(t)-(1+d)|t|^{\alpha}}\right\\},$

where for real $a,b$ and $\eta>0$ $[a,b]_{\eta}=[a,b]\cap\eta\mathbb{Z}$. Set
in the following $[a,b]_{0}:=[a,b]$ for $a<b$.

The following theorem provide us an upper bound for the difference between a
Piterbarg constant and the corresponding truncated Piterbarg constant:

######  Theorem 2.2.

It holds with some uniform in $\delta\geq 0$ constant $\mathcal{C}$ and all
sufficiently large $T$, that

$\displaystyle\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K})-\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K};T)\leq
e^{-\mathcal{C}T^{\alpha}}.$

The results above imply the following approach for approximation of
$\mathcal{P}_{\alpha}(d,\mathcal{K})$ for $d>0$:

  * 1)

For small $\delta>0$ take $S_{\delta}=(-\ln\delta)^{2/\alpha}$ and approximate
$\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K};S_{\delta})$ by
$\widehat{\mathcal{P}_{\alpha}^{\delta}}(d,\mathcal{K};S_{\delta})$ obtained
by Monte-Carlo simulations.

  * 2)

Theorem 2.1 and Theorem 2.2 imply that
$\mathcal{P}_{\alpha}(d,\mathcal{K})-\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K};S_{\delta})\leq\mathcal{C}\delta^{\alpha/2}(-\ln\delta)^{1/2}$
uniformly for sufficiently small $\delta>0$, hence we can approximate
$\mathcal{P}_{\alpha}(d,\mathcal{K})$ by
$\widehat{\mathcal{P}_{\alpha}^{\delta}}(d,\mathcal{K};S_{\delta})$ with an
error term not exceeding $\mathcal{C}\delta^{\alpha/2}(-\ln\delta)^{1/2}$
combined with statistical error from estimation of
$\widehat{\mathcal{P}_{\alpha}^{\delta}}(d,\mathcal{K};S_{\delta})$.

Independence of the increments of Brownian motion allows us to deduce more
precise bounds for
$\mathcal{P}_{1}(d,\mathcal{K})-\mathcal{P}_{1}^{\delta}(d,\mathcal{K})$ than
in Theorem 2.1. Namely,

######  Theorem 2.3.

For $d>0$ as $\delta\to 0$ it holds, that

$\displaystyle\mathcal{P}_{1}(d,\mathcal{K})-\mathcal{P}_{1}^{\delta}(d,\mathcal{K})\sim-\frac{\zeta(1/2)}{\sqrt{\pi}}\sqrt{\delta}\cdot\mathcal{P}_{1}^{\delta}(d,\mathcal{K}),$

where $\zeta$ is the Euler-Riemann zeta function and
$-\frac{\zeta(1/2)}{\sqrt{\pi}}>0$.

It is interesting, that the constant $-\zeta(1/2)/\sqrt{\pi}$ appears above,
the same constant plays the same role in the difference between the classical
Pickands constants, see [3]. This constant appears in many problems concerning
the difference between supremum of BM on a continuous and discrete grids, see
[10].

##  3\. Proofs

Define for $d>0$ and $\alpha\in(0,2)$

(1) $\displaystyle
Z_{\alpha}(t)=\sqrt{2}B_{\alpha}(t)-(d+1)|t|^{\alpha},\qquad t\in\mathbb{R}.$

Before giving proofs we present and prove several lemmas needed for the proofs
of the main results.

######  Lemma 3.1.

For all $d>0$ there exists $C_{1},C_{2}\in\mathbb{R}$ such that

$\displaystyle\mathbb{P}\left\\{\sup\limits_{t\in[0,\infty)}e^{Z_{\alpha}(t)}>x\right\\}\sim
C_{1}(\ln x)^{C_{2}}x^{-1-d},\qquad x\to\infty.$

###### Proof of Lemma 3.1.

By the self-similarity of fBM, for $x>1$ we have

$\displaystyle\mathbb{P}\left\\{\sup\limits_{t\in[0,\infty)}e^{Z_{\alpha}(t)}>x\right\\}$
$\displaystyle=$ $\displaystyle\mathbb{P}\left\\{\exists
t\in[0,\infty):\sqrt{2}B_{\alpha}(t)-(d+1)t^{\alpha}>\ln x\right\\}$
$\displaystyle=$ $\displaystyle\mathbb{P}\left\\{\exists
t\in[0,\infty):\frac{\sqrt{2}B_{\alpha}(t)}{(d+1)t^{\alpha}+1}>\sqrt{\ln
x}\right\\}=:\mathbb{P}\left\\{\exists t\in[0,\infty):V(t)>\sqrt{\ln
x}\right\\}.$

The variance of $V(t)$ for $t\geq 0$ achieves its unique maximum at
$t_{0}=(\frac{1}{\sqrt{d+1}})^{2/\alpha}$ and
$\operatorname*{\mathbb{V}ar}\\{V(t_{0})\\}=\frac{1}{2(d+1)},$ hence applying
[16, Theorem 10.1] we find that there exist some $C_{1}>0$ and
$C_{2}\in\mathbb{R}$ such that
$\mathbb{P}\left\\{\sup_{t\in[0,\infty)}e^{Z_{\alpha}(t)}>x\right\\}\sim
C_{1}(\ln x)^{C_{2}}x^{-1-d}$, as $x\to\infty$. ∎

######  Lemma 3.2.

For any $p,T>0$ and $\alpha\in(0,2)$ for sufficiently small $\delta>0$ with
some $\mathcal{C}>0$ that does not depend on $\delta$ it holds, that

$\displaystyle\mathbb{E}\left\\{\sup\limits_{t,s\in[0,T],|t-s|\leq\delta}|B_{\alpha}(t)-B_{\alpha}(s)|^{p}\right\\}^{1/p}\leq\mathcal{C}\delta^{\alpha/2}\sqrt{\ln(|\delta/T|)}.$

###### Proof of Lemma 3.2.

We have by Theorem 4.2 in [6] with some non-negative random variable
$\mathcal{A}$ that does not depend on $\delta$ and has all finite moments that

$\displaystyle\mathbb{E}\left\\{\sup\limits_{t,s\in[0,T],|t-s|\leq\delta}|B_{\alpha}(t)-B_{\alpha}(s)|^{p}\right\\}$
$\displaystyle=$ $\displaystyle T^{\frac{\alpha
p}{2}}\mathbb{E}\left\\{\sup\limits_{t,s\in[0,1],|t-s|\leq\frac{\delta}{T}}|B_{\alpha}(t)-B_{\alpha}(s)|^{p}\right\\}$
$\displaystyle\leq$ $\displaystyle T^{\frac{\alpha
p}{2}}\mathbb{E}\left\\{\sup\limits_{t,s\in[0,1],|t-s|\leq\frac{\delta}{T}}\left(\mathcal{A}|t-s|^{\alpha/2}\sqrt{|\ln(|t-s|/T)|}\right)^{p}\right\\}$
$\displaystyle=$ $\displaystyle T^{\frac{\alpha
p}{2}}(\frac{\delta}{T})^{\alpha
p/2}\sqrt{|\ln(\delta/T)|}^{p}\mathbb{E}\left\\{\mathcal{A}^{p}\right\\}$
$\displaystyle=$ $\displaystyle C\delta^{\alpha
p/2}\sqrt{|\ln(\delta/T)|}^{p},$

and the claim follows. ∎

Now we are ready to prove the main results.

###### Proof of Theorem 2.1.

We have by Theorem 2.2 with $T_{\delta}=(-\ln\delta)^{2/\alpha}$ that

$\displaystyle\mathcal{P}_{\alpha}(d,\mathcal{K})-\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K})$
$\displaystyle\leq$
$\displaystyle\mathcal{P}_{\alpha}(d,\mathcal{K};T_{\delta})-\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K};T_{\delta})+o(\delta^{\alpha/2}).$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\sup\limits_{t\in\mathcal{K}\cap[-T_{\delta},T_{\delta}]}e^{Z_{\alpha}(t)}-\sup\limits_{t\in\mathcal{K}\cap[-T_{\delta},T_{\delta}]_{\delta}}e^{Z_{\alpha}(t)}\right\\}+o(\delta^{\alpha/2})$
$\displaystyle\leq$ $\displaystyle
2\mathbb{E}\left\\{\sup\limits_{t\in[0,T_{\delta}]}e^{Z_{\alpha}(t)}-\sup\limits_{t\in[0,T_{\delta}]_{\delta}}e^{Z_{\alpha}(t)}\right\\}+o(\delta^{\alpha/2}).$

Note that for any $y\leq x$ it holds, that $e^{x}-e^{y}\leq(x-y)e^{x}$.
Implementing this inequality we find that, for $s,t\in[0,T_{\delta}]$,

$\displaystyle\left|e^{Z_{\alpha}(t)}-e^{Z_{\alpha}(s)}\right|$
$\displaystyle\leq$ $\displaystyle
e^{\max(Z_{\alpha}(t),Z_{\alpha}(s))}|Z_{\alpha}(t)-Z_{\alpha}(s)|$
$\displaystyle\leq$ $\displaystyle
e^{\max\limits_{w\in[0,\infty)}Z_{\alpha}(w)}\left|\sqrt{2}(B_{\alpha}(t)-B_{\alpha}(s))-(t^{\alpha}-s^{\alpha})\right|.$

By the lines above we obtain

$\displaystyle\mathbb{E}\left\\{\sup\limits_{t\in[0,T_{\delta}]}e^{Z_{\alpha}(t)}-\sup\limits_{t\in[0,T_{\delta}]_{\delta}}e^{Z_{\alpha}(t)}\right\\}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left\\{\sup\limits_{t,s\in[0,T_{\delta}],|t-s|\leq\delta}|e^{Z_{\alpha}(t)}-e^{Z_{\alpha}(t)}|\right\\}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left\\{\sup\limits_{t,s\in[0,T_{\delta}],|t-s|\leq\delta}e^{\max\limits_{w\in[0,\infty)}Z_{\alpha}(w)}\left|\sqrt{2}(B_{\alpha}(t)-B_{\alpha}(s))-(t^{\alpha}-s^{\alpha})\right|\right\\}$
$\displaystyle\leq$
$\displaystyle\sqrt{2}\mathbb{E}\left\\{\sup\limits_{t,s\in[0,T_{\delta}],|t-s|\leq\delta}e^{\max\limits_{w\in[0,\infty)}Z_{\alpha}(w)}\left|B_{\alpha}(t)-B_{\alpha}(s)\right|\right\\}+\mathbb{E}\left\\{e^{\max\limits_{w\in[0,\infty)}Z_{\alpha}(w)}\right\\}\sup\limits_{t,s\in[0,T_{\delta}],|t-s|\leq\delta}\left|t^{\alpha}-s^{\alpha}\right|$
$\displaystyle\leq$
$\displaystyle\sqrt{2}\mathbb{E}\left\\{e^{q\max\limits_{w\in[0,\infty)}Z_{\alpha}(w)}\right\\}^{1/q}\mathbb{E}\left\\{\sup\limits_{t,s\in[0,T_{\delta}],|t-s|\leq\delta}\left|B_{\alpha}(t)-B_{\alpha}(s)\right|^{p}\right\\}^{1/p}+o(\delta^{\alpha/2}),$

where $1/q+1/p=1$ and $q$ is chosen such that
$\mathbb{E}\left\\{e^{q\max\limits_{w\in[0,\infty)}Z_{\alpha}(w)}\right\\}<\infty$,
this is possible to do by Theorem 3.1. Next by Lemma 3.2 and lines above we
have

$\displaystyle\mathcal{P}_{\alpha}(d,\mathcal{K})-\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K})\leq\mathcal{C}\delta^{\alpha/2}\sqrt{\ln(|\delta/T_{\delta}|)}+o(\delta^{\alpha/2})$

and the claim follows. ∎

###### Proof of Theorem 2.2.

We have

$\displaystyle\mathcal{P}_{\alpha}^{\delta}(a)-\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K};T)=\mathbb{E}\left\\{\sup\limits_{t\in\mathcal{K}\cap(-\infty,\infty)_{\delta}}e^{Z_{\alpha}(t)}-\sup\limits_{t\in\mathcal{K}\cap[-T,T]_{\delta}}e^{Z_{\alpha}(t)}\right\\}\leq\mathbb{E}\left\\{\sup\limits_{t\in\mathbb{R}}e^{Z_{\alpha}(t)}\mathbb{I}(M_{T})\right\\},$

where $M_{T}=\\{\omega\in\Omega:Z_{\alpha}(\omega,t)>0\text{ for some
}t\in(-\infty,-T]\cap[T,\infty)\\}$ with $\Omega$ being the general
probability space. We have by Borell-TIS inequality and the symmetry

$\displaystyle\mathbb{P}\left\\{M_{T}\right\\}\leq 2\mathbb{P}\left\\{\exists
t\geq T:\sqrt{2}B_{\alpha}(t)-t^{\alpha}>0\right\\}=2\mathbb{P}\left\\{\exists
t\geq T:\frac{\sqrt{2}B_{\alpha}(t)}{t^{\alpha/2}}>t^{\alpha/2}\right\\}\leq
e^{-CT^{\alpha}}.$

By Lemma 3.1 $\sup\limits_{t\in\mathbb{R}}e^{Z_{\alpha}(t)}$ has the finite
moment of order $1+d/2$ and hence applying Hölder inequality we have

$\displaystyle\mathbb{E}\left\\{\sup\limits_{t\in\mathbb{R}}e^{Z_{\alpha}(t)}\mathbb{I}(M_{T})\right\\}\leq
C_{1}\mathbb{P}\left\\{M_{T}\right\\}^{C_{2}}\leq e^{-CT^{\alpha}}$

and the claim follows.∎

###### Proof of Theorem 2.3.

Let $Z_{\alpha}(t)=\sqrt{2}B_{\alpha}(t)-(1+a)|t|^{\alpha}$ be defined as in
(1). We will show the statement only for $\mathcal{K}=(-\infty,\infty)$; the
proof for $\mathcal{K}=[0,\infty)$ will be analogous (but slightly simpler).
Fix $a>0$ for $J\subseteq\mathbb{R}$, we define

$M^{J}=\sup_{t\in J}Z_{\alpha}(t),\quad M^{J}_{\delta}=\sup_{t\in
J\cap\delta\mathbb{Z}}Z_{\alpha}(t),\quad\Delta^{J}_{\delta}:=\frac{M^{J}-M^{J}_{\delta}}{\delta^{1/2}}.$

For breviety we write $M:=M^{\mathbb{R}},M_{\delta}:=M^{\mathbb{R}}_{\delta}$,
and $\Delta_{\delta}:=\Delta^{\mathbb{R}}_{\delta}$. Then,

(2)
$\mathcal{P}_{\alpha}(d,\mathcal{K})-\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K})=\mathbb{E}\left\\{e^{M}-e^{M_{\delta}}\right\\}.$

Using mean value theorem and noticing that $M_{\delta}\leq M$ a.s., we obtain
the following bounds

(3) $\mathbb{E}\left\\{(M-M_{\delta})\cdot
e^{M_{\delta}}\right\\}\leq\mathcal{P}_{\alpha}(d,\mathcal{K})-\mathcal{P}_{\alpha}^{\delta}(d,\mathcal{K})\leq\mathbb{E}\left\\{(M-M_{\delta})\cdot
e^{M}\right\\}.$

Till the end, we want to show that

(4) $\lim_{\delta\to 0}\mathbb{E}\left\\{\Delta_{\delta}\cdot
e^{M_{\delta}}\right\\}=\lim_{\delta\to
0}\mathbb{E}\left\\{\Delta_{\delta}\cdot e^{M}\right\\}=\lim_{\delta\to
0}\mathbb{E}\left\\{\Delta_{\delta}\right\\}\cdot\mathcal{P}_{\alpha}(d,\mathcal{K})=-\frac{\zeta(1/2)}{\sqrt{\pi}}\cdot\mathcal{P}_{\alpha}(d,\mathcal{K}),$

which, in combination with (3), will conclude the proof. We will show (4) in
three steps.

Step 1. Show that there exists a random variable $D$ such that
$\mathbb{E}\left\\{D\right\\}=-\zeta\big{(}\tfrac{\alpha-1}{\alpha}\big{)}\mathbb{E}\left\\{\widehat{X}_{1}^{+}\right\\}$
and such that for any continuity set $A$ of $D$ and an arbitrary event
$B\in\mathcal{F}$ of positive probability, where $\mathcal{F}$ is the borel
$\sigma$-algebra, under which $Z$ is adapted to the usual fitration, for
$J=\mathbb{R}$, as $\delta\downarrow 0$

(5) $\mathbb{P}\left\\{\Delta_{\delta}^{J}\in
A,B\right\\}\to\mathbb{P}\left\\{D\in A\right\\}\mathbb{P}\left\\{B\right\\}.$

The statement (5) for $J=\mathbb{R}$ implies that
$\Delta_{\delta}\overset{d}{\to}D$ and, intuitively, that the sequence is
‘asymptotically independent’ of the process
$\\{Z_{\alpha}(t)\\}_{t\in\mathbb{R}}$.

Now, [12, Thm. 4] implies that (5) holds for $J=[0,T)$ as well as $J=(-T,0]$
for arbitrary $T\in(0,\infty)$.

Now, let $\tau$ and $\tau_{\delta}$ be the (almost surely unique) time epochs
of the all-time supremum of $Z$ over $\mathbb{R}$ and over
$\delta\mathbb{\mathbb{Z}}$ respectively, i.e. $Z_{\alpha}(\tau)=M$ and
$Z_{\alpha}(\tau_{\delta})=M_{\delta}$. For any $T>0$, let $C_{T}^{+}$ be the
event that $\\{\tau\in[0,T),\tau_{\delta}\in[0,T)\\}$ and similarly, let
$C_{T}^{-}:=\\{\tau\in(-T,0],\tau_{\delta}\in(-T,0]\\}$. We have

$\displaystyle\mathbb{P}\left\\{\Delta_{\delta}\in
A,B\right\\}=\mathbb{P}\left\\{\Delta_{\delta}\in
A,B,C_{T}^{-}\right\\}+\mathbb{P}\left\\{\Delta_{\delta}\in
A,B,C_{T}^{+}\right\\}+\mathbb{P}\left\\{\Delta_{\delta}\in A,B,(C_{T}^{-}\cup
C_{T}^{+})^{\prime}\right\\}$
$\displaystyle=\mathbb{P}\left\\{\Delta^{[0,T)}_{\delta}\in
A,B,C_{T}^{+}\right\\}+\mathbb{P}\left\\{\Delta_{\delta}^{(-T,0]}\in
A,B,C_{T}^{-}\right\\}+\mathbb{P}\left\\{\Delta_{\delta}\in A,B,(C_{T}^{-}\cup
C_{T}^{+})^{\prime}\right\\},$

which implies that

$\displaystyle\Big{|}\mathbb{P}\left\\{\Delta_{\delta}\in
A,B\right\\}-\big{(}\mathbb{P}\left\\{\Delta^{[0,T)}_{\delta}\in
A,B,C_{T}^{+}\right\\}+\mathbb{P}\left\\{\Delta^{(-T,0]}_{\delta}\in
A,B,C_{T}^{-}\right\\}\big{)}\Big{|}\leq\mathbb{P}\left\\{(C_{T}^{-}\cup
C_{T}^{+})^{\prime}\right\\}.$

We now pass $\delta\to 0$, use (5) with $B:=B\cap C_{T}^{+}$ and $B:=B\cap
C_{T}^{-}$, and obtain

$\displaystyle\Big{|}\mathbb{P}\left\\{\Delta_{\delta}\in
A,B\right\\}-\mathbb{P}\left\\{D\in
A\right\\}\big{(}\mathbb{P}\left\\{B,C_{T}^{+}\right\\}$
$\displaystyle+\mathbb{P}\left\\{B,C_{T}^{-}\right\\}\big{)}\Big{|}\leq\mathbb{P}\left\\{(C_{T}^{-}\cup
C_{T}^{+})^{\prime}\right\\}.$

We pass $T\to\infty$ and notice that $C_{T}^{+}\cap C_{T}^{-}=\emptyset$ and
$\mathbb{P}\left\\{C_{T}^{+}\cup C_{T}^{-}\right\\}\to 1$, as $T\to\infty$,
which shows (5) for $J=\mathbb{R}$. Finally, we note that
$\mathbb{E}\left\\{D\right\\}=-\frac{\zeta(1/2)}{\sqrt{\pi}}$, which is
established in [2, Corollary 4].

Step 2. Show that the family of random variables $\Delta_{\delta}\cdot
e^{M_{\delta}}$, indexed by $\delta$, is uniformly integrable for $\delta$
small enough.

According to de la Vallée-Poussin Theorem, it is enough to establish that
there exists $\varepsilon>0$ such that the sequence
$\mathbb{E}\left\\{(\Delta_{\delta}\cdot
e^{M_{\delta}})^{1+\varepsilon}\right\\}$ is bounded for all $\delta$ small
enough. Due to Hölder’s inequality, with $p,q\in[1,\infty)$ satisfying
$1/p+1/q=1$ we obtain

(6) $\mathbb{E}\left\\{(\Delta_{\delta}\cdot
e^{M_{\delta}})^{1+\varepsilon}\right\\}\leq\left(\mathbb{E}\left\\{\Delta_{\delta}^{p(1+\varepsilon)}\right\\}\right)^{1/p}\cdot\left(\mathbb{E}\left\\{e^{q(1+\varepsilon)M_{\delta}}\right\\}\right)^{1/q}.$

We now establish that for any $0<p<\alpha$, the $p$th moment of
$\Delta_{\delta}$ is bounded for all $\delta$ small enough. Since
$\Delta_{\delta}\leq\Delta_{\delta}^{[0,\infty)}+\Delta_{\delta}^{(-\infty,0]}$,
then for any $p>0$ we have

$\displaystyle\mathbb{E}\left\\{\Delta_{\delta}\right\\}^{p}$
$\displaystyle\leq\max\\{1,2^{p-1}\\}\cdot\left(\mathbb{E}\left\\{\Delta^{[0,\infty)}_{\delta}\right\\}^{p}+\mathbb{E}\left\\{\Delta^{(-\infty,0]}_{\delta}\right\\}^{p}\right),$

where we used the fact that $(x+y)^{p}\leq\max\\{1,2^{p-1}\\}(x^{p}+y^{p})$
for all $x,y>0$. The boundedness of
$\mathbb{E}\left\\{\Delta_{\delta}\right\\}^{p}$ readily follows from the
boundedness of $\mathbb{E}\left\\{\Delta^{[0,\infty)}_{\delta}\right\\}^{p}$
and $\mathbb{E}\left\\{\Delta^{(-\infty,0]}_{\delta}\right\\}^{p}$, which are
established in [2, Theorem 2]. Moreover, since $M$ is exponentially
distributed with mean $(1+a)^{-1}$, then $\mathbb{E}\left\\{e^{\alpha
M}\right\\}<\infty$ for any $\alpha<1+a$. So, in (6), if we take $q>1$ and
$\varepsilon>0$ small enough such that $q(1+\varepsilon)<1+a$, then
$\mathbb{E}\left\\{e^{q(1+\varepsilon)M_{\delta}}\right\\}\leq\mathbb{E}\left\\{e^{q(1+\varepsilon)M}\right\\}$
will be bounded, as $\delta\to 0$.

Step 3. Since $e^{M_{\delta}}\to e^{M}$ a.s., and the limit is
$\mathcal{F}$-measurable, then due to the mixing condition in (5), the pair
$(\Delta_{\delta},e^{M_{\delta}})$ converges jointly in distribution to
$(D,e^{M})$, where $D$ is independent of $e^{M}$. Now, from the continuous
mapping theorem we find that $\Delta_{\delta}\cdot
e^{M_{\delta}}\overset{d}{\to}D\cdot e^{M}$. Finally, (4) follows from the
uniform integrability of the family $\Delta_{\delta}\cdot e^{M_{\delta}}$. ∎

## Acknowledgements

We would like to thank Prof. Enkelejd Hashorva and Prof. Krzysztof Dȩbicki for
fruitful discussions. Krzysztof Bisewski’s research was funded by SNSF Grant
200021-196888. G. Jasnovidov was supported by the Ministry of Science and
Higher Education of the Russian Federation, agreement 075-15-2019-1620 date
08/11/2019 and 075-15-2022-289 date 06/04/2022.

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