EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2015-222 LHCb-PAPER-2015-037 September 2, 2015

Measurement of forward ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
production cross-sections in $pp$ collisions at
$\sqrt{s}=13\mathrm{\,Te\kern-2.07413ptV}$

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

The production of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons in
proton-proton collisions at a centre-of-mass energy of
$\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$ is studied with the LHCb detector.
Cross-section measurements are performed as a function of the transverse
momentum $p_{\rm T}$ and the rapidity $y$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson in the region
$\mbox{$p_{\rm T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$,
for both prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons from $b$-hadron decays.
The production cross-sections integrated over the kinematic coverage are
$15.30\pm 0.03\pm 0.86{\rm\,\upmu b}$ for prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $2.34\pm 0.01\pm
0.13{\rm\,\upmu b}$ for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from
$b$-hadron decays, assuming zero polarization of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. The first uncertainties
are statistical and the second systematic. The cross-section reported for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons from $b$-hadron decays
is used to extrapolate to a total $b\bar{b}$ cross-section. The ratios of the
cross-sections with respect to $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ are
also determined.

Submitted to JHEP

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

##  1 Introduction

The study of heavy quarkonium production in $pp$ collisions provides important
information on both the perturbative and the non-perturbative regimes of
quantum chromodynamics (QCD). Heavy quarkonium production can be described in
two stages. The first is the short-distance production of a heavy quark pair,
$Q\overline{Q}$, which can be described perturbatively and the second is the
non-perturbative hadronisation of the heavy quark pair into quarkonium state,
such as the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ meson. The non-
perturbative part cannot yet be determined reliably, and must be determined
using experimental results. After almost forty years of theoretical and
experimental efforts, the hadronic production of quarkonia is still not fully
understood. In the colour singlet model (CSM) [Carlson:1976cd,
Donnachie:1976ue, Ellis:1976fj, Fritzsch:1977ay, Gluck:1977zm, Chang:1979nn,
Baier:1981uk], the intermediate $Q\overline{Q}$ state is assumed to be
colourless, and has the same $J^{PC}$ quantum numbers as the final-state
quarkonium. In the non-relativistic QCD (NRQCD) approach [Bodwin:1994jh,
Cho:1995vh, Cho:1995ce], all viable colour and spin-parity quantum numbers are
allowed for the intermediate $Q\overline{Q}$ state, and each configuration is
assigned a probability to transform into the specific quarkonium state. The
transition probabilities are described by universal long-distance matrix
elements (LDME) determined from experimental data.

In $pp$ collisions, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons can
be produced directly from hard collisions of partons, through the feed-down of
excited charmonium states, or via decays of $b$-flavoured hadrons. The first
two sources are collectively referred to as prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production, while the third
process is referred to in the following as
“${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$”.

The ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ differential production
cross-section has been measured with LHC data in $pp$ collisions at centre-of-
mass energies of $2.76\mathrm{\,Te\kern-1.00006ptV}$ [Abelev:2012kr, LHCb-
PAPER-2012-039], $7\mathrm{\,Te\kern-1.00006ptV}$ [LHCb-PAPER-2011-003,
Khachatryan:2010yr, Aad:2011sp, Aamodt:2011gj, Chatrchyan:2011kc], and
$8\mathrm{\,Te\kern-1.00006ptV}$ [LHCb-PAPER-2013-016]. Next-to-leading order
CSM calculations [Campbell:2007ws, Lansberg:2011hi] give a better description
of the experimental data than leading order (LO) calculations at high
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse momentum
($\mbox{$p_{\rm T}$}\\!/c>M({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}})$).
However, CSM calculations still underestimate the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production cross-section. On
the other hand, NRQCD calculations with LDME determined from CDF data
[Abe:1992ww] describe well the $p_{\rm T}$ dependence of the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-section at both the
Tevatron [Cacciari:1995yt, Braaten:1994vv] and the LHC experiments [Ma:2010jj,
Gong:2008ft, Butenschoen:2010rq]. The measured
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$ production cross-
section and its dependence on $p_{\rm T}$ at the LHC are in good agreement
with predictions from fixed order plus next-to-leading logarithms (FONLL)
[Cacciari:1998it] calculations.

The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarisation was measured
by the ALICE [Abelev:2011md], CMS [Chatrchyan:2013cla] and LHCb [LHCb-
PAPER-2013-008] collaborations in $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The measurements by ALICE were
performed for inclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson
production, while CMS and LHCb disentangled prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$ mesons. Next-to-
leading order CSM calculations predict a large longitudinal polarisation of
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson [Gong:2008sn], while
NRQCD calculations predict a large ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ transverse polarisation at high $p_{\rm T}$ [Beneke:1995yb,
Chao:2012iv, Gong:2012ug, Butenschoen:2012px]. Neither prediction is supported
by experimental results [Abulencia:2007us, Abazov:2008aa, Abelev:2011md,
Chatrchyan:2013cla, Chatrchyan:2012woa, LHCb-PAPER-2013-008, LHCb-
PAPER-2013-067].

This paper reports ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-
section measurements in $pp$ collisions at
$\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$ in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ kinematic range $\mbox{$p_{\rm
T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$, using
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decaying to
${\mu^{+}}{\mu^{-}}$ final states. Under the assumption of zero
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarisation (see Sec. 5 for
detailed discussion), the following quantities are measured: the double
differential cross-sections as a function of $p_{\rm T}$ and $y$; the
integrated production cross-sections for prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$; the
${{b}{\overline{b}}}$ production cross-section; and the ratio of cross-
sections with respect to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
cross-sections in $pp$ collisions at
$\sqrt{s}=8~{}\mathrm{\,Te\kern-1.00006ptV}$ previously measured by LHCb
[LHCb-PAPER-2013-016].

##  2 The LHCb detector and data set

The data used in this analysis come from $pp$ collisions at
$\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$, collected by the LHCb detector in
July 2015, with an average of 1.1 visible interactions per bunch crossing and
correspond to an integrated luminosity of $3.05\pm 0.12\mbox{\,pb}^{-1}$. The
LHCb detector [Alves:2008zz, LHCb-DP-2014-002] 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
$4{\rm\,Tm}$, 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${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. 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
[LHCb-DP-2012-002].

The online event selection consists of a hardware stage, based on information
from the calorimeter and muon systems, followed by a software stage, which
performs ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate
reconstruction. The hardware trigger selects events with at least one muon
candidate with transverse momentum $\mbox{$p_{\rm
T}$}>0.9{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In the first stage of the
software trigger, two muon tracks with $\mbox{$p_{\rm
T}$}>500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ are required to form a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate with invariant mass
$M({\mu^{+}\mu^{-}})>2.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. In the
second stage, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates with
good vertex fit quality and invariant mass within
$150{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known value [PDG2014] are
selected.

This analysis benefits from a new scheme for the LHCb software trigger
introduced for LHC Run 2. Alignment and calibration is performed in near real-
time [Dujany:2017839] and updated constants are made available for the
trigger. The same alignment and calibration information is propagated to the
offline reconstruction, to ensure consistent and high-quality particle
identification information for the trigger and offline. The larger timing
budget available in the trigger with respect to LHC Run 1 also results in the
convergence of the online and offline track reconstruction, such that offline
performance is achieved in the trigger. The identical performance of the
online and offline reconstruction achieved in this way offers the opportunity
to perform physics analyses directly using candidates reconstructed in the
trigger [LHCb-DP-2012-004]. The storage of only the triggered candidates
enables a reduction in the event size by an order of magnitude. The analysis
described in this paper uses the online reconstruction for the first time in
LHCb, and is checked the against the standard offline reconstruction chain.

Simulated samples are used to evaluate the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ detection efficiency. In the
simulation, $pp$ collisions are generated using Pythia 6 [pythia6.4] with a
specific LHCb configuration [LHCb-PROC-2010-056]. Decays of hadronic particles
are described by EvtGen [Lange:2001uf], in which final-state radiation is
generated using Photos [Golonka:2005pn]. The prompt charmonium production is
simulated in Pythia with contributions from both the leading order colour-
singlet and colour-octet contributions [LHCb-PROC-2010-056, Bargiotti:2007zz],
and the charmonium is generated unpolarised. The interaction of the generated
particles with the detector, and its response, are simulated using the Geant4
toolkit [Allison:2006ve, *Agostinelli:2002hh] as described in Ref. [LHCb-
PROC-2011-006].

##  3 Selection of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates

The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates are selected in
the second step of the software trigger. Each event is required to have at
least one primary vertex (PV) reconstructed from at least four tracks found by
the vertex detector. For events with multiple PVs, the PV which has the
smallest $\chi^{2}_{\mathrm{IP}}$ with respect to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidate is chosen. The
$\chi^{2}_{\mathrm{IP}}$ is defined as the change of the primary vertex fit
quality when the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ meson is
excluded from the PV fit. Each identified muon track is required to have
$\mbox{$p_{\rm T}$}>0.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$p>3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and to have a good quality track
fit. The muon tracks of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidate must form a good quality two-track vertex. Duplicate tracks created
by the reconstruction are suppressed to the level of $0.5\times 10^{-3}$.

The reconstructed vertex of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
mesons originating from $b$-hadron decays tends to be separated from the PVs,
and thus these can be distinguished from prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons by exploiting the pseudo
decay time defined as

$t_{z}=\frac{\left(z_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}-z_{\mathrm{PV}}\right)\times
M_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}}{p_{z}},$ (1)

where $z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}-z_{\mathrm{PV}}$ is
the distance along the beam axis between the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay vertex and the PV,
$p_{z}$ is the $z$-component of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ momentum, and $M_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ the
known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [PDG2014].

The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates with
$\left|t_{z}\right|<10{\rm\,ps}$, corresponding to less than 7 times the
$b$-hadron lifetime, are selected for the fits to the $t_{z}$ distribution. To
further select good ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates,
the uncertainty on $t_{z}$, which is propagated from the uncertainties
provided by the track reconstruction, is required to be less than
$0.3{\rm\,ps}$.

##  4 Cross-section determination

The double differential ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
production cross-section in each kinematic bin of $p_{\rm T}$ and $y$ is
defined as

$\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}y\mathrm{d}\mbox{$p_{\rm
T}$}}=\frac{N({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}})}{L_{\text{int}}\times{\varepsilon_{\rm
tot}}\times{\cal B}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}\mu^{-}})\times\Delta y\times\Delta\mbox{$p_{\rm
T}$}},$ (2)

where $N({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}\mu^{-}})$ is the yield of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$ signal mesons,
${\varepsilon_{\rm tot}}$ is the total detection efficiency in the given
kinematic bin, $L_{\text{int}}$ is the integrated luminosity, ${\cal
B}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}\mu^{-}})=(5.961\pm 0.033)\%$ [PDG2014] is the
branching ratio of the decay ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}\mu^{-}}$ and $\Delta\mbox{$p_{\rm
T}$}=1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\Delta y=0.5$ are the bin
widths. The measurements are restricted to $\mbox{$p_{\rm
T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ due to limited data at higher
transverse momenta.

The absolute luminosity is determined from the beam profiles and beam
currents. The beam profiles and their overlap integral are measured using a
beam-gas imaging method [Barschel:1693671, LHCb-PAPER-2014-047], where neon is
injected into the beam vacuum around the interaction point. The beam currents
are measured by LHC instruments, which determine the bunch population
fractions and the total beam intensity. Furthermore, information on beam-gas
interactions not originating from nominally filled bunch slots is used to
determine the charge fraction not participating in bunch collisions. The
integrated luminosity for this analysis is calibrated using the number of
visible $pp$ interactions measured during the beam-gas imaging runs and during
the runs used for this analysis. From this procedure, the integrated
luminosity is measured to be $3.05\pm 0.12\mbox{\,pb}^{-1}$.

The efficiency ${\varepsilon_{\rm tot}}$ is determined as the product of the
reconstruction and selection efficiencies, muon identification efficiency and
trigger efficiency, and is calculated using simulated samples in each
$(\mbox{$p_{\rm T}$},y)$ bin, independently for prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$. The track
reconstruction and the muon identification efficiency are corrected using
data-driven techniques, while the trigger efficiencies are also validated
using data, as explained in Sec. 5.

The yield of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal events,
both from prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$, is determined from a
two-dimensional unbinned maximum likelihood fit to the invariant mass and
pseudo decay time of the candidates, performed independently for each
$(\mbox{$p_{\rm T}$},y)$ bin. The invariant mass distribution of the signal is
described by the sum of two Crystal Ball (CB) functions [Skwarnicki:1986xj]
with a common mean value and different widths. The parameters of the power law
tails, the relative fractions and the difference between the widths of the two
CB functions are fixed to values obtained from the simulation, leaving the
mean and one width of the CB function as free parameters. The combinatorial
background is described by an exponential distribution.

The fraction of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$
candidates, $F_{b}$, is determined from the fit to the $t_{z}$ distribution.
The $t_{z}$ distribution of prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ is described by a Dirac $\delta$ function at $t_{z}=0$, and that of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$ by an exponential
decay function, which are both convolved with a double-Gaussian resolution
function. A ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate can also
be associated to a wrong PV, resulting in a long tail component in the $t_{z}$
distribution. The fraction of the tail is below 0.5% of all events. Its shape
is modelled from data by calculating $t_{z}$ with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate from a given event
and the closest PV in the next event of the sample. The background $t_{z}$
distribution is parametrised with an empirical function based on the observed
shape of the $t_{z}$ distribution in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass sidebands. The background
comes from muons of semileptonic $b$\- and $c$-hadron decays and from pions
and kaons decaying in-flight. It is parametrised as the sum of a Dirac
$\delta$ function and five exponential functions, three for positive $t_{z}$
and two for negative $t_{z}$. The function is convolved with a double-Gaussian
resolution function similar to that used for the signal, but with different
parameters. All parameters of the background $t_{z}$ distribution are fixed to
values determined from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass
sidebands independently in each $(\mbox{$p_{\rm T}$},y)$ bin.

The total ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal yield
determined from the fit is about one million events. An example for one
$(\mbox{$p_{\rm T}$},y)$ bin of the invariant mass and the pseudo decay time
distributions is shown in Fig. 1 with the one-dimensional projections of the
fit result superimposed.

Figure 1: Invariant mass (left) and pseudo decay time (right) distributions
for the kinematic bin $2<\mbox{$p_{\rm
T}$}<3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $3.0<y<3.5$, with fit results
superimposed. The solid (red) line is the total fit function, the shaded
(green) area corresponds to the background component. The prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ contribution is shown in cross-
hatched area (blue), ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$
in a solid (black) line and the tail contribution due to the association of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ with the wrong PV is shown in
full filled (magenta) area. The tail contribution is not visible in the
invariant mass plot.

##  5 Systematic uncertainties

Systematic uncertainties, most of which apply to both prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\text{-from-}b$ mesons, are
summarised in Table 1 and described below.

Table 1: Relative systematic uncertainties (in %) on the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-section measurements. The uncertainty from the $t_{z}$ fit only affects ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$-from-$b$ mesons. Most of the uncertainties are fully correlated between bins, with the exception of the $p_{\rm T}$, $y$ spectrum dependence and the simulation statistics, which are considered uncorrelated. Source | Systematic uncertainty (%)
---|---
Luminosity | 3.9
Hardware trigger | $0.1-5.9$
Software trigger | 1.5
Muon ID | 1.8
Tracking | $1.1-3.4$
Radiative tail | 1.0
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ vertex fit | 0.4
Signal mass shape | 1.0
$\mathcal{B}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}})$ | 0.6
$p_{\rm T}$, $y$ spectrum | $0.1-5.0$
Simulation statistics | $0.3-5.0$
$t_{z}$ fit (${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$-from-$b$ only) | 0.1

The uncertainty related to the modelling of the signal mass shape is studied
by replacing the nominal model with a Hypatia function [Santos:2013gra], which
takes into account the mass uncertainty distribution. The relative difference
of the signal yield is about $1.0\%$, which is taken as a fully correlated
systematic uncertainty in each bin.

Due to the presence of bremsstrahlung in the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}}$
decay, a fraction of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events
fall outside the mass window used in this analysis. The efficiency of the mass
window selection is determined from simulation, and based on a detailed
comparison between the radiative tails in simulation and data, a value of
$1.0\%$ of the yield is assigned as the systematic uncertainty.

To calibrate the muon identification efficiency determined from simulation,
the single-track muon identification efficiency is measured with a
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}}$
data sample using a tag-and-probe method. In this method, the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates are reconstructed
with only one track identified as a muon (“tag”). The single muon
identification efficiency is measured as the probability of the other track
(“probe”) to be identified as a muon, in bins of momentum, $p_{\mu}$, and
pseudorapidity, $\eta_{\mu}$ of the probe track. The single-track muon
identification efficiency obtained in data is weighted with the
$(p_{\mu},\eta_{\mu})$ distribution of the muons from
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons in simulation. The
resulting efficiency is divided by that determined directly from simulation,
giving $1.050\pm 0.017$, which is used to correct the muon identification
efficiency in each $(\mbox{$p_{\rm T}$},y)$ bin of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. The uncertainty on the
correction factor, limited by the calibration sample size in data, is used to
obtain the systematic uncertainty. The choice of binning scheme in
$(p_{\mu},\eta_{\mu})$ is another source of systematic uncertainty and is
evaluated by choosing alternative binning schemes. In total the systematic
uncertainty on the cross-sections due to the muon identification is $1.8\%$.

The tracking efficiency is studied with a data-driven tag-and-probe approach
with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays using partially
reconstructed tracks, where one muon track is fully reconstructed as the tag
track, and the probe track is reconstructed using only specific sub-detectors
[LHCb-DP-2013-002]. The simulated sample is weighted to agree in event
multiplicity with the data sample. A systematic uncertainty of the efficiency
of $0.4\%$ per muon track is assigned to account for a different event
multiplicity between data and simulation. The tracking efficiency is
determined to be the fraction of ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}}$ decays where the probe track can be
matched to a fully reconstructed track. The ratio of the resulting tracking
efficiency between data and simulation is used to weight the simulation sample
according to pseudorapidity and momentum of muons to obtain an efficiency
correction in each $(\mbox{$p_{\rm T}$},y)$ bin of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. In total the correction
factor ranges from $0.94$ to $1.04$, depending on the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $(\mbox{$p_{\rm T}$},y)$ bin.
The uncertainty on the muon track efficiency correction factor is limited by
the size of the calibration data sample, and is propagated into the systematic
uncertainty on the cross-section measurements. In total, the systematic
uncertainty on the cross-sections related to the tracking efficiency is in the
range of $1-3\%$, depending on the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ $(\mbox{$p_{\rm T}$},y)$ bin.

The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ vertex fit quality
requirement leads to an uncertainty of $0.4\%$ for the selection efficiency,
which is determined by comparing the distributions of the vertex fit quality
between data and simulation.

The trigger efficiency obtained from simulation is cross-checked using data-
driven methods using a fully reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ sample. For the hardware
trigger, a tag-and-probe method is used to evaluate the single track muon
trigger efficiency in bins of muon transverse momentum, $\mbox{$p_{\rm
T}$}_{\mu}$, and pseudorapidity, $\eta_{\mu}$, for both simulation and data.
In this procedure, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidate is required to pass both steps of the software trigger. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ trigger efficiency is
calculated by weighting $\mbox{$p_{\rm T}$}_{\mu}$ and $\eta_{\mu}$ of the
muons with the single-muon track trigger efficiency obtained from data and
simulation, and their relative difference is quoted as a systematic
uncertainty. For most of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
$(\mbox{$p_{\rm T}$},y)$ bins, the systematic effect on the cross-section is
found to be below $1.0\%$, but in two bins an uncertainty of 5.9% is found.
The software trigger efficiency is determined using a subset of events that
would have triggered if the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
signals were excluded [LHCb-DP-2012-004]. The efficiency is computed in each
$(\mbox{$p_{\rm T}$},y)$ bin for data and simulation, and the relative
difference between data and simulation of about $1.5\%$ is taken as a
systematic uncertainty.

The possible discrepancy between the $p_{\rm T}$ and $y$ distributions of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons in data and simulation
for each bin is studied by reweighting the distribution in simulation to that
in data. The relative difference between the efficiency after reweighting and
the nominal efficiency is taken as a systematic uncertainty and found to be in
the range $0.1\%-5\%$, depending on the $(\mbox{$p_{\rm T}$},y)$ bin of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, where the largest value
corresponds to the bins at the LHCb acceptance boundary.

The uncertainty associated with the luminosity determination is $3.9\%$, and
the branching fraction uncertainty of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{+}\mu^{-}}$
decay is $0.6\%$. The limited size of the simulated sample in each bin leads
to an uncertainty between $0.3\%$ and $5\%$, which is less than half of the
data statistical uncertainty in each bin.

The detection efficiency is dependent on the polarisation of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. Previous measurements by
CDF [Abulencia:2007us] in $p\overline{p}$ collisions at
$1.96\mathrm{\,Te\kern-1.00006ptV}$, ALICE [Abelev:2011md], CMS
[Chatrchyan:2013cla] and LHCb [LHCb-PAPER-2013-008] in $pp$ collisions at
$7\mathrm{\,Te\kern-1.00006ptV}$, showed that the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarisation in hadron
collisions is small. The LHCb experiment studied
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarisation in the helicity
frame [Jacob:1959at] and measured the longitudinal polarisation parameter
$\lambda_{\theta}$ [LHCb-PAPER-2013-008] to be on average $-0.145\pm 0.027$ in
the range $2<\mbox{$p_{\rm T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y<4.5$. If the longitudinal polarisation is assumed to be $-20$%, the
measured ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-section would
decrease by values between $0.7\%$ and $6.2\%$ depending on the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $(\mbox{$p_{\rm T}$},y)$ bin,
with an average value of $2.5\%$, which is smaller than the total systematic
uncertainty. The efficiency changes in different $p_{\rm T}$ and $y$ bins
obtained under this assumption are discussed in the appendix. Therefore, since
no polarisation measurement has yet been made for data collected at
$\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$, the polarisation is assumed to be
zero, and no corresponding systematic uncertainty is quoted on the cross-
section related to this effect.

There are sources of systematic uncertainties that are related to the $t_{z}$
fit, that affect only ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$
decays and are negligible for prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ production. The modelling of the $t_{z}$ resolution is studied by
adding a third Gaussian to the nominal resolution model. The variation in
$F_{b}$ is found to be negligible. Background modelling is tested using the
sPlot [Pivk:2004ty] method to extract the background $t_{z}$ distribution and
the relative variation in $F_{b}$ is taken as a systematic uncertainty. The
impact of the choice of $t_{z}$ parametrisation for the long tail component is
studied using an exponential function with equal magnitude for positive and
negative slopes and the relative difference of $F_{b}$ is taken as a
systematic uncertainty. The total relative systematic uncertainty on the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\text{-from-}b$ cross-section
related to the $t_{z}$ fit is 0.1%.

##  6 Results

The measured double differential cross-sections for prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$ mesons, assuming no
polarisation, are shown in Figs. LABEL:fig:ResultPromptJpsi and
LABEL:fig:ResultJpsiFromB, and given in Tables 6 and
LABEL:tab:ResultJpsiFromB. The cross-sections for prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-from-$b$ mesons in the
acceptance $\mbox{$p_{\rm T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y<4.5$, integrated over all $(\mbox{$p_{\rm T}$},y)$ bins, are:

$\displaystyle\sigma(\mathrm{prompt}~{}{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\mbox{$p_{\rm T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},2.0<y<4.5)$
$\displaystyle=$ $\displaystyle 15.30\pm 0.03\pm 0.86{\rm\,\upmu b},$
$\displaystyle\sigma({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\text{-from-}b,\mbox{$p_{\rm
T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},2.0<y<4.5)$ $\displaystyle=$
$\displaystyle\phantom{0}2.34\pm 0.01\pm 0.13{\rm\,\upmu b},$

where the first uncertainties are statistical and the second systematic.

Table 2:  Double differential production cross-section in
$\mathrm{nb}/(\\!{\mathrm{\,Ge\kern-0.90005ptV\\!/}c})$ for prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons in bins of
($\mbox{$p_{\rm T}$},y$). The first uncertainties are statistical, the second
are the correlated systematic uncertainties shared between bins and the last
are the uncorrelated systematic uncertainties.

$\mbox{$p_{\rm T}$}\,[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}]$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$
---|---|---|---
$0$ | $-$ | $1$ |  | $1059\,\,\,$ | $\,\pm\,$ | $16\,\,\,$ | $\,\pm\,$ | $46\,\,\,$ | $\,\pm\,$ | $32\,\,\,$ | $947\,\,\,$ | $\,\pm\,$ | $9\,\,\,$ | $\,\pm\,$ | $38\,\,\,$ | $\,\pm\,$ | $8\,\,\,$ | $861\,\,\,$ | $\,\pm\,$ | $8\,\,\,$ | $\,\pm\,$ | $39\,\,\,$ | $\,\pm\,$ | $5\,\,\,$
$1$ | $-$ | $2$ |  | $2079\,\,\,$ | $\,\pm\,$ | $22\,\,\,$ | $\,\pm\,$ | $89\,\,\,$ | $\,\pm\,$ | $17\,\,\,$ | $1878\,\,\,$ | $\,\pm\,$ | $12\,\,\,$ | $\,\pm\,$ | $75\,\,\,$ | $\,\pm\,$ | $11\,\,\,$ | $1683\,\,\,$ | $\,\pm\,$ | $10\,\,\,$ | $\,\pm\,$ | $77\,\,\,$ | $\,\pm\,$ | $6\,\,\,$
$2$ | $-$ | $3$ |  | $1863\,\,\,$ | $\,\pm\,$ | $18\,\,\,$ | $\,\pm\,$ | $78\,\,\,$ | $\,\pm\,$ | $17\,\,\,$ | $1659\,\,\,$ | $\,\pm\,$ | $11\,\,\,$ | $\,\pm\,$ | $66\,\,\,$ | $\,\pm\,$ | $13\,\,\,$ | $1431\,\,\,$ | $\,\pm\,$ | $9\,\,\,$ | $\,\pm\,$ | $63\,\,\,$ | $\,\pm\,$ | $5\,\,\,$
$3$ | $-$ | $4$ |  | $1218\,\,\,$ | $\,\pm\,$ | $13\,\,\,$ | $\,\pm\,$ | $51\,\,\,$ | $\,\pm\,$ | $14\,\,\,$ | $1062\,\,\,$ | $\,\pm\,$ | $7\,\,\,$ | $\,\pm\,$ | $43\,\,\,$ | $\,\pm\,$ | $6\,\,\,$ | $928\,\,\,$ | $\,\pm\,$ | $6\,\,\,$ | $\,\pm\,$ | $39\,\,\,$ | $\,\pm\,$ | $3\,\,\,$
$4$ | $-$ | $5$ |  | $755\,\,\,$ | $\,\pm\,$ | $9\,\,\,$ | $\,\pm\,$ | $31\,\,\,$ | $\,\pm\,$ | $9\,\,\,$ | $620\,\,\,$ | $\,\pm\,$ | $5\,\,\,$ | $\,\pm\,$ | $25\,\,\,$ | $\,\pm\,$ | $4\,\,\,$ | $527\,\,\,$ | $\,\pm\,$ | $4\,\,\,$ | $\,\pm\,$ | $22\,\,\,$ | $\,\pm\,$ | $2\,\,\,$
$5$ | $-$ | $6$ |  | $419\,\,\,$ | $\,\pm\,$ | $6\,\,\,$ | $\,\pm\,$ | $17\,\,\,$ | $\,\pm\,$ | $5\,\,\,$ | $351\,\,\,$ | $\,\pm\,$ | $3\,\,\,$ | $\,\pm\,$ | $14\,\,\,$ | $\,\pm\,$ | $2\,\,\,$ | $288\,\,\,$ | $\,\pm\,$ | $3\,\,\,$ | $\,\pm\,$ | $12\,\,\,$ | $\,\pm\,$ | $1\,\,\,$
$6$ | $-$ | $7$ |  | $236.7\,$ | $\,\pm\,$ | $3.7\,$ | $\,\pm\,$ | $9.6\,$ | $\,\pm\,$ | $7.4$ | $188.4\,$ | $\,\pm\,$ | $2.2\,$ | $\,\pm\,$ | $7.5\,$ | $\,\pm\,$ | $1.3$ | $157.8\,$ | $\,\pm\,$ | $1.9\,$ | $\,\pm\,$ | $6.4\,$ | $\,\pm\,$ | $0.9$
$7$ | $-$ | $8$ |  | $130.2\,$ | $\,\pm\,$ | $2.5\,$ | $\,\pm\,$ | $5.3\,$ | $\,\pm\,$ | $3.8$ | $104.0\,$ | $\,\pm\,$ | $1.6\,$ | $\,\pm\,$ | $4.2\,$ | $\,\pm\,$ | $0.8$ | $85.6\,$ | $\,\pm\,$ | $1.4\,$ | $\,\pm\,$ | $3.4\,$ | $\,\pm\,$ | $0.6$
$8$ | $-$ | $9$ |  | $76.0\,$ | $\,\pm\,$ | $1.8\,$ | $\,\pm\,$ | $3.1\,$ | $\,\pm\,$ | $1.4$ | $58.2\,$ | $\,\pm\,$ | $1.2\,$ | $\,\pm\,$ | $2.3\,$ | $\,\pm\,$ | $0.6$ | $48.3\,$ | $\,\pm\,$ | $1.0\,$ | $\,\pm\,$ | $1.9\,$ | $\,\pm\,$ | $0.4$
$9$ | $-$ | $10$ |  | $47.0\,$ | $\,\pm\,$ | $1.3\,$ | $\,\pm\,$ | $1.9\,$ | $\,\pm\,$ | $1.1$ | $36.0\,$ | $\,\pm\,$ | $0.9\,$ | $\,\pm\,$ | $1.4\,$ | $\,\pm\,$ | $0.4$ | $27.3\,$ | $\,\pm\,$ | $0.7\,$ | $\,\pm\,$ | $1.1\,$ | $\,\pm\,$ | $0.3$
$10$ | $-$ | $11$ |  | $28.7\,$ | $\,\pm\,$ | $1.0\,$ | $\,\pm\,$ | $1.1\,$ | $\,\pm\,$ | $0.6$ | $20.6\,$ | $\,\pm\,$ | $0.7\,$ | $\,\pm\,$ | $0.8\,$ | $\,\pm\,$ | $0.4$ | $16.9\,$ | $\,\pm\,$ | $0.6\,$ | $\,\pm\,$ | $0.7\,$ | $\,\pm\,$ | $0.2$
$11$ | $-$ | $12$ |  | $16.9\,$ | $\,\pm\,$ | $0.8\,$ | $\,\pm\,$ | $0.7\,$ | $\,\pm\,$ | $0.5$ | $12.7\,$ | $\,\pm\,$ | $0.5\,$ | $\,\pm\,$ | $0.5\,$ | $\,\pm\,$ | $0.2$ | $11.0\,$ | $\,\pm\,$ | $0.5\,$ | $\,\pm\,$ | $0.4\,$ | $\,\pm\,$ | $0.2$
$12$ | $-$ | $13$ |  | $11.5\,$ | $\,\pm\,$ | $0.6\,$ | $\,\pm\,$ | $0.5\,$ | $\,\pm\,$ | $0.4$ | $7.5\,$ | $\,\pm\,$ | $0.4\,$ | $\,\pm\,$ | $0.3\,$ | $\,\pm\,$ | $0.2$ | $6.6\,$ | $\,\pm\,$ | $0.4\,$ | $\,\pm\,$ | $0.3\,$ | $\,\pm\,$ | $0.1$
$13$ | $-$ | $14$ |  | $7.2\,$ | $\,\pm\,$ | $0.5\,$ | $\,\pm\,$ | $0.3\,$ | $\,\pm\,$ | $0.3$ | $5.5\,$ | $\,\pm\,$ | $0.3\,$ | $\,\pm\,$ | $0.2\,$ | $\,\pm\,$ | $0.1$ | $3.9\,$ | $\,\pm\,$ | $0.3\,$ | $\,\pm\,$ | $0.2\,$ | $\,\pm\,$ | $0.1$
|  |  |  | $3.5<y<4.0$ | $4.0<y<4.5$ |  |  |  |  |  |  |
$0$ | $-$ | $1$ |  | $807\,\,\,$ | $\,\pm\,$ | $7\,\,\,$ | $\,\pm\,$ | $45\,\,\,$ | $\,\pm\,$ | $5\,\,\,$ | $724\,\,\,$ | $\,\pm\,$ | $8\,\,\,$ | $\,\pm\,$ | $48\,\,\,$ | $\,\pm\,$ | $9\,\,\,$ |  |  |  |  |  |  |
$1$ | $-$ | $2$ |  | $1503\,\,\,$ | $\,\pm\,$ | $9\,\,\,$ | $\,\pm\,$ | $87\,\,\,$ | $\,\pm\,$ | $6\,\,\,$ | $1333\,\,\,$ | $\,\pm\,$ | $11\,\,\,$ | $\,\pm\,$ | $95\,\,\,$ | $\,\pm\,$ | $17\,\,\,$ |  |  |  |  |  |  |
$2$ | $-$ | $3$ |  | $1239\,\,\,$ | $\,\pm\,$ | $8\,\,\,$ | $\,\pm\,$ | $69\,\,\,$ | $\,\pm\,$ | $5\,\,\,$ | $1010\,\,\,$ | $\,\pm\,$ | $10\,\,\,$ | $\,\pm\,$ | $73\,\,\,$ | $\,\pm\,$ | $14\,\,\,$ |  |  |  |  |  |  |
$3$ | $-$ | $4$ |  | $772\,\,\,$ | $\,\pm\,$ | $6\,\,\,$ | $\,\pm\,$ | $40\,\,\,$ | $\,\pm\,$ | $3\,\,\,$ | $624\,\,\,$ | $\,\pm\,$ | $8\,\,\,$ | $\,\pm\,$ | $42\,\,\,$ | $\,\pm\,$ | $11\,\,\,$ |  |  |  |  |  |  |
$4$ | $-$ | $5$ |  | $429\,\,\,$ | $\,\pm\,$ | $4\,\,\,$ | $\,\pm\,$ | $20\,\,\,$ | $\,\pm\,$ | $2\,\,\,$ | $322\,\,\,$ | $\,\pm\,$ | $5\,\,\,$ | $\,\pm\,$ | $19\,\,\,$ | $\,\pm\,$ | $7\,\,\,$ |  |  |  |  |  |  |
$5$ | $-$ | $6$ |  | $237\,\,\,$ | $\,\pm\,$ | $3\,\,\,$ | $\,\pm\,$ | $11\,\,\,$ | $\,\pm\,$ | $1\,\,\,$ | $169\,\,\,$ | $\,\pm\,$ | $3\,\,\,$ | $\,\pm\,$ | $9\,\,\,$ | $\,\pm\,$ | $3\,\,\,$ |  |  |  |  |  |  |
$6$ | $-$ | $7$ |  | $119.6\,$ | $\,\pm\,$ | $1.7\,$ | $\,\pm\,$ | $5.2\,$ | $\,\pm\,$ | $0.8$ | $87.3\,$ | $\,\pm\,$ | $2.1\,$ | $\,\pm\,$ | $4.5\,$ | $\,\pm\,$ | $1.0$ |  |  |  |  |  |  |
$7$ | $-$ | $8$ |  | $64.6\,$ | $\,\pm\,$ | $1.2\,$ | $\,\pm\,$ | $2.8\,$ | $\,\pm\,$ | $0.5$ | $44.8\,$ | $\,\pm\,$ | $1.4\,$ | $\,\pm\,$ | $2.2\,$ | $\,\pm\,$ | $1.3$ |  |  |  |  |  |  |
$8$ | $-$ | $9$ |  | $36.3\,$ | $\,\pm\,$ | $0.9\,$ | $\,\pm\,$ | $1.5\,$ | $\,\pm\,$ | $0.3$ | $24.4\,$ | $\,\pm\,$ | $1.0\,$ | $\,\pm\,$ | $1.2\,$ | $\,\pm\,$ | $1.0$ |  |  |  |  |  |  |
$9$ | $-$ | $10$ |  | $20.1\,$ | $\,\pm\,$ | $0.7\,$ | $\,\pm\,$ | $0.8\,$ | $\,\pm\,$ | $0.2$ | $13.1\,$ | $\,\pm\,$ | $0.7\,$ | $\,\pm\,$ | $0.6\,$ | $\,\pm\,$ | $0.4$ |  |  |  |  |  |  |
$10$ | $-$ | $11$ |  | $11.6\,$ | $\,\pm\,$ | $0.5\,$ | $\,\pm\,$ | $0.5\,$ | $\,\pm\,$ | $0.2$ | $7.4\,$ | $\,\pm\,$ | $0.5\,$ | $\,\pm\,$ | $0.3\,$ | $\,\pm\,$ | $0.4$ |  |  |  |  |  |  |
$11$ | $-$ | $12$ |  | $6.8\,$ | $\,\pm\,$ | $0.4\,$ | $\,\pm\,$ | $0.3\,$ | $\,\pm\,$ | $0.1$ | $4.4\,$ | $\,\pm\,$ | $0.4\,$ | $\,\pm\,$ | $0.2\,$ | $\,\pm\,$ | $0.2$ |  |  |  |  |  |  |
$12$ | $-$ | $13$ |  | $4.4\,$ | $\,\pm\,$ | $0.3\,$ | $\,\pm\,$ | $0.2\,$ | $\,\pm\,$ | $0.1$ | $2.7\,$ | $\,\pm\,$ | $0.2\,$ | $\,\pm\,$ | $0.1\,$ | $\,\pm\,$ | $0.1$ |  |  |  |  |  |  |
$13$ | $-$ | $14$ |  | $2.5\,$ | $\,\pm\,$ | $0.2\,$ | $\,\pm\,$ | $0.1\,$ | $\,\pm\,$ | $0.1$ | $2.1\,$ | $\,\pm\,$ | $0.2\,$ | $\,\pm\,$ | $0.1\,$ | $\,\pm\,$ | $0.1$ |  |  |  |  |  |  |

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


ITEP-TH-31/18

Toda flow and intersections of Bruhat cells

Yu.B. Chernyakov111Institute for Theoretical and Experimental Physics,
Bolshaya Cheremushkinskaya, 25, 117218 Moscow, Russia.,222Joint Institute for
Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, 141980 Dubna,
Moscow region, Russia<EMAIL_ADDRESS>
G.I Sharygin11footnotemark: 1,22footnotemark: 2,333Lomonosov Moscow State
University, Faculty of Mechanics and Mathematics, GSP-1, 1 Leninskiye Gory,
Main Building, 119991 Moscow, Russia<EMAIL_ADDRESS>
A.S. Sorin22footnotemark: 2,444National Research Nuclear University MEPhI
(Moscow Engineering Physics Institute), Kashirskoye shosse 31, 115409 Moscow,
Russia,555Dubna State University, 141980 Dubna (Moscow region), Russia.,
<EMAIL_ADDRESS>
D.V.Talalaev11footnotemark: 1,33footnotemark: 3,666Centre of integrable
systems, P. G. Demidov Yaroslavl State University, 150003, Yaroslavl, Russian
Federation, 14 Sovetskaya str<EMAIL_ADDRESS>
###### Abstract

In this short note we show that the Bruhat cells in real normal forms of
semisimple Lie algebras enjoy the same property as their complex analogs: for
any two elements $w,w^{\prime}$ in the Weyl group $W(\mathfrak{g})$, the
corresponding real Bruhat cell $X_{w}$ intersects with the dual Bruhat cell
$Y_{w^{\prime}}$ iff $w\prec w^{\prime}$ in the weak Bruhat order on
$W(\mathfrak{g})$. Here $\mathfrak{g}$ is a normal real form of a semisimple
complex Lie algebra $\mathfrak{g}_{\mathbb{C}}$. Our reasoning is based on the
properties of the Toda flows, rather than on the analysis of the Weyl group
action and geometric considerations.

##  1 Introduction and the statement of result

Schubert cells in standard flag manifolds and their generalizations to
arbitrary Lie groups, Bruhat cells, play important role in algebraic geometry
and representation theory. In complex case their structure is well understood
and has been extensively studied in the last 40 years.

These cells are open dense subsets in algebraic subvarieties in flag spaces
(Schubert varieties), enumerated by the elements of the corresponding Weyl
group. Namely, the Schubert cell $X_{w}^{\mathbb{C}}$, corresponding to $w\in
W(\mathfrak{g}_{\mathbb{C}})$ is equal to the positive Borel subgroup
$B^{+}_{\mathbb{C}}$ orbit of the element $[w]\in
Fl(\mathfrak{g}_{\mathbb{C}})=G_{\mathbb{C}}/B^{+}_{\mathbb{C}}$. Here $[w]$
is the point in $Fl(\mathfrak{g}_{\mathbb{C}})$, represented by the element
$\tilde{w}$ in the normalizer $N(\mathfrak{h})$ of Cartan subalgebra,
corresponding to $w\in W(\mathfrak{g}_{\mathbb{C}})$. Similarly dual Schubert
cell $Y_{w}^{\mathbb{C}}$, corresponding to $w$ is the orbit
$B^{-}_{\mathbb{C}}\cdot[w]$ of the negative Borel subgroup.

It turns out that the properties of the Bruhat cells are closely related with
the combinatoric of the corresponding Weyl group, in particular with the
strong Bruhat order on it. Recall, that Bruhat order in a Coxeter group $W$
with the set $S=\\{\sigma_{1},\dots,\sigma_{N}\\}$ of generators is the unique
partial order on $W$ determined by the following rule: we say that $w$
precedes $w^{\prime}$, if a minimal word $s(w^{\prime})$ (in letters
$\sigma_{k}$), representing $w^{\prime}$ has the form
$\sigma_{i_{1}}\cdot\dots\cdot\sigma_{i_{k}}\cdot s(w)$ (where $s(w)$ is a
minimal word for $w$); length of these words are called lengths of the
elements and denoted $l(w),\,l(w^{\prime})$ respectively.

Now using complex geometry and algebraic groups theory one can show that
$X^{\mathbb{C}}_{w}\subseteq\overline{X}^{\mathbb{C}}_{w^{\prime}}$ iff
$w\prec w^{\prime}$ in Bruhat order (here $\overline{X}$ denotes the closure
of $X$). Besides this, one can also show that $Y^{\mathbb{C}}_{w}\bigcap
X^{\mathbb{C}}_{w^{\prime}}\neq\emptyset$ iff $w^{\prime}\prec w$ in Bruhat
order (see [1, 2]). Moreover, in the latter case the intersection is always
transversal and the (complex) dimension of this variety is equal to the
difference $l(w)-l(w^{\prime})$, in particular, the complex dimensions of the
cells are equal to $l(w)$ and
$\dim_{\mathbb{C}}Fl(\mathfrak{g}_{\mathbb{C}})-l(w^{\prime})$ respectively
(see the cited papers).

It turned out that the proofs of similar properties for real Bruhat cells,
though mutely implied by many authors, have not been accurately written down
yet (or at least we couldn’t find a proper reference). The modest purpose of
the present note is to fill this regrettable gap, at least for the normal real
forms and to show the close relation of the Toda system with the geometry of
Bruhat cells. So let $\mathfrak{g}$ be a real Lie algebra, whose
complexification is equal to $\mathfrak{g}_{\mathbb{C}}$ (i.e. $\mathfrak{g}$
is a real form of $\mathfrak{g}_{\mathbb{C}}$); let $G$ be the corresponding
real Lie group. For any such real form of $\mathfrak{g}_{\mathbb{C}}$, one has
the following Cartan decomposition:

$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p},$

where $\mathfrak{k}$ is the Lie algebra of the maximal compact subgroup, and
$\mathfrak{p}$ is the algebraic complement of $\mathfrak{k}$ on which the
Killing form of $\mathfrak{g}$ is positive-definite.

Alternatively (and maybe more properly) one can define the Cartan
decomposition with the help of Cartan involution; namely let $\theta$ be the
Lie algebra automorphism $\theta:\mathfrak{g}\to\mathfrak{g}$, such that
$\theta^{2}=1$ and the quadratic form $B(X,\theta Y)$ where $B$ is the Killing
form, is positive definite. Then

$\mathfrak{k}=\\{X\in\mathfrak{g}\mid\theta(X)=-X\\},\
\mathfrak{p}=\\{Y\in\mathfrak{p}\mid\theta(Y)=Y\\}.$

The Cartan decomposition of a real form $\mathfrak{g}$ is unique (up to an
isomorphism) and one has

$[\mathfrak{k},\mathfrak{k}]\subseteq\mathfrak{k},\ \
[\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{k},\ \
[\mathfrak{k},\mathfrak{p}]\subseteq\mathfrak{p}.$

Let $\mathfrak{a}$ be a maximal (commutative) Lie subalgebra in
$\mathfrak{p}$; then the real form $\mathfrak{g}$ is called normal, or non-
split, if $\mathfrak{a}$ coincides with a maximal commutative subalgebra in
the whole $\mathfrak{g}$.

In this case we shall denote this commutative subalgebra by $\mathfrak{h}$ and
call it Cartan subalgebra. The normal real form enjoys most of the properties
of the complex algebra: one can determine the root system with respect to
$\mathfrak{h}$ and decompose $\mathfrak{g}$ into root subspaces:

$\mathfrak{g}=\mathfrak{h}\oplus\sum_{\alpha\in\Phi}\mathfrak{g}_{\alpha},$

where $\alpha\in\Phi\subset\mathfrak{h}^{*}$ are roots and
$\mathfrak{g}_{\alpha}$ are the root subspaces, i.e.

$\mathfrak{g}_{\alpha}=\\{X\in\mathfrak{g}\mid[H,X]=\alpha(H)X,\
H\in\mathfrak{h}\\}.$

In the case of normal real form these subspaces are $1$-dimensional and fixing
a basis in $\mathfrak{h}$ so that one can tell which roots are positive and
which are negative, one has the following decompositions of $\mathfrak{k}$ and
$\mathfrak{p}$ in the corresponding Cartan decomposition:

$\displaystyle\mathfrak{k}$
$\displaystyle=\sum_{\alpha\in\Phi_{+}}\mathbb{R}(e_{\alpha}+e_{-\alpha}),$
(1) $\displaystyle\mathfrak{p}$
$\displaystyle=\mathfrak{h}\oplus\sum_{\alpha\in\Phi_{+}}\mathbb{R}(e_{\alpha}-e_{-\alpha}),$

where $\Phi_{+}$ is the set of positive roots and $e_{\alpha},\,e_{-\alpha}$
are basis vectors in $\mathfrak{g}_{\alpha},\,\mathfrak{g}_{-\alpha}$ such
that $e_{-\alpha}=\theta(e_{\alpha})$. It is easy to see that in this case the
Cartan decomposition of the corresponding complex Lie algebra
$\mathfrak{g}_{\mathbb{C}}$ is given by the complexification of the
corresponding elements of $\mathfrak{g}$.

The Weyl group of $\mathfrak{g}$, which coincides with
$\mathfrak{g}_{\mathbb{C}}$ can now be defined as the factorgroup
$W(\mathfrak{g})=N_{K}(\mathfrak{h})/Z_{K}(\mathfrak{h})$, where
$N_{K}(\mathfrak{h})$ and $Z_{K}(\mathfrak{h})$ are the normalizer and the
centralizer of $\mathfrak{h}$ in the maximal compact subgroup $K$ of $G$,
corresponding to $\mathfrak{k}$. It follows from the maximality of
$\mathfrak{h}$ that these groups are discrete, and that $Z_{K}(\mathfrak{h})$
is normal subgroup in it.

On the other hand, one can define real Bruhat cells $X_{w}$ and dual real
Bruhat cells $Y_{w}$ in $Fl(G)=G/B^{+}=K/Z_{K}(\mathfrak{h})$ (where $B^{+}$
is the real Borel subgroup) as the orbits of $[w]$ with respect to $B^{+}$ and
$B^{-}$. In what follows, we shall show that the following is true:

######  Proposition 1.1.

The intersection $X_{w}\bigcap Y_{w^{\prime}}$ is nonempty, iff
$w^{\prime}\prec w$ in Bruhat order; moreover, if it is not empty, its (real)
dimension is equal to the difference $l(w)-l(w^{\prime})$.

In addition to filling the existing gap (we found a good explanation of this
fact only in the case of $G=SL_{n}(\mathbb{R})$, where it follows from the
geometric description of Schubert cells in terms of matrix ranks, see e.g.
[2]), the purpose of this article is to demonstrate the connection of this
theory with the full symmetric Toda flow. Namely, we are going to derive the
result we need from the general properties of the Toda flow, and not vice-
versa: in our previous work we used the properties of Schubert cells in the
standard flag variety (i.e. the one associated with $SL_{n}(\mathbb{R})$) in
order to show that the phase portrait of this dynamical system is completely
determined by Bruhat order ([3]).

Also observe that in our following papers ([4, 5]) we used explicit
computations to show that similar statements hold for all real groups of rank
$2$ and for the non-split real form, $SO(2,4)$. In the latter case the role of
Weyl group is played by the group
$W(\mathfrak{g},\mathfrak{k})=N_{K}(\mathfrak{a})/Z_{K}(\mathfrak{a})$, where
$N_{K},\,Z_{K}$ are the (non-discrete) normalizer and centralizer of
$\mathfrak{a}$ in $K$; one can show, that $W(\mathfrak{g},\mathfrak{k})$ is
again Coxeter group, thus we can introduce Bruhat order on it. Flag variety is
replaced by $K/Z_{K}(\mathfrak{a})$ and the (dual) Bruhat cells are defined as
orbits of Borel subgroups in flag varieties, similarly to the usual case.
However, in this case the dimensions of the root spaces (with respect to
$\mathfrak{a}$) are not necessarily equal to $1$; this is manifested in the
dimensions of the intersections of the cells. Below we shall describe a
possible generalization of our main result to this case.

##  2 Full symmetric Toda flow and its properies

In this section we recall few properties of the full symmetric Toda system on
Lie algebras and flag spaces. Namely, this is the dynamical system on
$\mathfrak{p}$, defined as follows (we follow the exposition from the papers
by de Mari, Pedroni [6] and Bloch, Gekhtman [8]): it is the Hamiltonian system
with respect to a Poisson structure, pulled to $\mathfrak{p}$ from
$(\mathfrak{b}^{+})^{*}$ via the Killing form, and with the Hamiltonian
$H(L)=\mathrm{Tr}_{\mathfrak{g}}(ad^{2}_{\mathfrak{g}}(L)),\
L\in\mathfrak{p}$.

However, for our purposes it is better to consider only the corresponding
Hamiltonian vector field $\tau$ on $\mathfrak{p}$; it can be written down in
the following commutator form:

$\tau(L)=[L,M(L)].$

Here for

$L=X+\sum_{\alpha\in\Phi_{+}}a_{\alpha}(e_{\alpha}+e_{-\alpha})\in\mathfrak{p},\
X\in\mathfrak{h},\ a_{\alpha}\in\mathbb{R}$

(see (1)) we put

$M(L)=\sum_{\alpha\in\Phi_{+}}a_{\alpha}(e_{\alpha}-e_{-\alpha})\in\mathfrak{k}.$

Thus the Toda system can be written in the following Lax form:

$\dot{L}=[L,M(L)],$ (2)

Below we shall refer to the map $M:\mathfrak{p}\to\mathfrak{k}$ as to the Toda
projector, and the system (2) as the full symmetric Toda system on
$\mathfrak{g}$, or by a slight abuse of terminology, just Toda system on
$\mathfrak{g}$ since it is clear that it depends only on the real form
$\mathfrak{g}$ (conventionally, the term Toda system is used to denote the
dynamical system, determined by (2) on the subspace in $\mathfrak{p}$, spanned
by $\mathfrak{h}$ and the simple root spaces).

It turns out, that the dynamics of the Toda system (2) is uniquely determined
by the following construction: it is known that almost all (i.e. up to a
measure $0$ subset) orbits of $K$ on $\mathfrak{p}$ pass through a unique (up
to the action of $N\in\mathfrak{h}$) element of $\mathfrak{h}$; on the other
hand, the vector field $\tau$ is evidently tangent to these orbits (because it
is determined by the adjoint action of $\mathfrak{k}$). So choose a generic
element $\Lambda\in\mathfrak{h}$, such that its centralizer coincides with $C$
and consider the vector field $T$ at a point $\Psi\in K$, given by

$T^{\Lambda}(\Psi)=-M(Ad_{\Psi}(\Lambda))\cdot\Psi,$

i.e. it is the right translation by $\Psi$ of the element
$M(Ad_{\Psi}(\Lambda))\in\mathfrak{k}$. Then this vector field gives a
dynamical system $\Psi=\Psi(t)$ on $K$, such that the solution of Toda system
on $\mathfrak{p}$ through a generic point $L=L(0)$ is given by the formula

$L(t)=Ad_{\Psi(t)}(\Lambda),$

where $\Lambda\in\mathfrak{h}$ is the eigenmatrix of $L$ (the one fixed above)
and $\Psi(t)$ is the trajectory of the field $T$ on $K$, such that
$Ad_{\Psi(0)}(\Lambda)=L$ 777It would probably be more accurate to consider
instead of the field $T$ on $K$ an analogous field $T^{\prime}$ on the flag
space; but in that case the formula for $L(t)$ would involve a choice of
section $F(G)\to K$..

###  2.1 Invariant surfaces of the Toda field

It is our purpose to study the behaviour of the vector field $\tau$ on
$\mathfrak{p}$ by studying the equivalent fields $T^{\Lambda}$ on $K$. It has
been studied by many authors in the last 25 years, so here we shall just give
references for the well-known facts and give brief proofs for those which seem
to be less known.

One can show (see [7] and [8]) that the vector field $T^{\Lambda}$ on $K$ is
gradient; the corresponding potential function $F=F_{\Lambda}:K\to\mathbb{R}$
is given by

$F_{\Lambda}(\Psi)=\mathrm{Tr}_{\mathfrak{g}}(ad_{\mathfrak{g}}(Ad_{\Psi}(\Lambda))ad_{\mathfrak{g}}(N)),$

for a suitable element $N\in\mathfrak{h}$. It turns out that $F_{\Lambda}$ is
a Morse function. The construction also involves a proper choice of the
Riemann structure $\langle,\rangle$ on $K$. We shall not use the exact form of
the metric and of the element $N$ here; so far it is enough to know that the
scalar product in question is left-invariant and that in basis
$e_{\alpha}-e_{-\alpha}$ on the Lie algebra $\mathfrak{k}$ it coincides with
the Killing form up to a scalar factors $j_{\alpha}$, look into [6, 3] for
details; as for $N$, it is uniquely defined up to the element, central in
$\mathfrak{p}$.

From the very definition of the field $T^{\Lambda}$ it follows that for a
generic $\Lambda$ the only singular points of this field coincide with the
group $N_{K}(\mathfrak{h})$, or, on the level of the flag space $Fl(G)$, with
the set of elements of $W(\mathfrak{g})$. Further, one can show that there
always exists a vast family of invariant (with respect to $T^{\Lambda}$)
curves in $K$, i.e. of $1$-dimensional invariant submanifolds of the Toda
flow. In fact, it is possible to give an explicit description of such curves.
To this end we let $w\in N_{K}(\mathfrak{h})\subset K$ be a point and let us
fix a root $\alpha\in\Phi_{+}$. Consider the smooth path
$\gamma_{w}=\gamma_{w,\alpha}$ through the point $w$ on $K$, given by

$\gamma_{w}(t)=\exp{(t(e_{\alpha}-e_{-\alpha}))}\cdot w.$

######  Proposition 2.1.

The path $\gamma_{w}$ is tangent to the vector field $T$ on $K$.

###### Proof.

Let us consider the path

$\gamma_{w,\alpha}(t)=\exp{(t(e_{\alpha}-e_{-\alpha}))}\cdot w,$

and the corresponding path in the Lie algebra

$L(t)=Ad_{\gamma_{w,\alpha}(t)}(\Lambda)=Ad_{\exp{(t(e_{\alpha}-e_{-\alpha}))}}(\Lambda_{w})$

where the element $\Lambda_{w}\in\mathfrak{h}$ is given by

$\Lambda_{w}=Ad_{w}(\Lambda).$

Since the action of $\mathfrak{h}$ on $e_{\pm\alpha}$ is diagonal, we can
prove by induction that $L(t)$ is an element in the subspace in
$\mathfrak{p}$, generated by $\mathfrak{h}$ and ${e}_{\alpha}+e_{-\alpha}$.
Hence the $M(L(t))$ is proportional to $e_{\alpha}-e_{-\alpha}.$ Let us
introduce a scalar function $k(t)$ by

$M(L(t))=k(t)(e_{\alpha}-e_{-\alpha}).$

Then the Toda field $T$ at the point $\gamma_{w}(t)$ is equal to
$k(t)(e_{\alpha}-e_{-\alpha})\cdot\gamma_{w}(t)$ and is proportional to the
tangent field to the trajectory $\gamma_{w,\alpha}(t)$ with coefficient
$k(t)$. ∎

In addition to these invariant curves, we shall need the following result,
which one can find, for instance in [9, 6]:  Bruhat cells are invariant
manifolds of this system. In fact, one can make this statement more precise,
bringing it to the following form.

For every $\Lambda\in\mathfrak{h}$ we obtain an ordering of the elements of
$W(\mathfrak{g})$ by the values of $F_{\Lambda}$; let $w\in W(\mathfrak{g})$
be the minimal point. Then for any $w^{\prime}$ we consider the shifted Bruhat
cell $X_{w^{\prime}}^{w}$ and dual shifted Bruhat cell $Y_{w^{\prime}}^{w}$,
i.e. the orbits $(B^{+})^{w}\cdot[w^{\prime}]$ and
$(B^{-})^{w}\cdot[w^{\prime}]$, where $(B^{\pm})^{w}=w\cdot B^{\pm}\cdot
w^{-1}$ and $[w^{\prime}]$ is the point in $Fl(G)$, corresponding to
$w^{\prime}$. Then the shifted and dual shifted Bruhat cells are invariant
submanifolds of $T^{\Lambda}$; moreover Bruhat cells are unstable and dual
Bruhat cells are stable submanifolds of this system. See the cited papers for
proof.

Before we proceed, let us make the following observation: Bruhat cells, dual
Bruhat cells and their intersections are invariant surfaces of vector fields
$T^{\lambda}$ for all $\Lambda\in\mathfrak{h}$. This follows directly from the
previously cited facts: just observe that if two vector fields are tangent to
a surface, then all their linear combinations also are and notice that the
fields $T^{\Lambda}$ are linear in $\Lambda$.

##  3 Proof of the main result

We begin with the following simple observation: if the intersection
$X_{w}\bigcap Y_{w^{\prime}}$ is nonempty, then the elements $w$ and
$w^{\prime}$ are comparable in Bruhat order so that $w^{\prime}\prec w$.
Indeed, real Bruhat cells lie inside complex ones, and this statement is true
in the complex case. Also observe, that the real dimension of the
intersections of the real Bruhat cells with dual real Bruhat cells does not
exceed $l(w)-l(w^{\prime})$, since this is the complex dimension of the
intersection of complex cells.

Further, suppose that $\Lambda\in\mathfrak{h}$ is such that the minimal point
is equal to the unit element $e\in W(\mathfrak{g})$ (this is possible, since
$F_{\Lambda}(\Psi\cdot w)=F_{Ad_{w}(\Lambda)}(\Psi)$, so if we choose
$\Lambda^{\prime}=Ad_{w}(\Lambda)$, where $w$ is the current minimal element,
then $e$ will be minimum of $F_{\Lambda^{\prime}}$). Then it follows from the
properties of Bruhat cells and Toda system, listed in the end of the previous
section, that if there exists a trajectory $k(t)$ of $T^{\Lambda}$, which
tends to $w$ when $t\to+\infty$ and to $w^{\prime}$ when $t\to-\infty$, then
$w^{\prime}\prec w$; indeed, in this case the corresponding Bruhat and dual
Bruhat cells intersect.

Consider now the paths $\gamma_{w,\alpha}$, corresponding to simple roots
$\alpha_{1},\dots,\alpha_{r}$. It is known from the classical Lie groups
theory, that there exists $t_{0}\in\mathbb{R}$ such that
$\exp(t_{0}(e_{\alpha}-e_{-\alpha}))=\sigma_{\alpha}$, where $\sigma_{\alpha}$
is the reflection of $\mathfrak{h}$, corresponding to $\alpha$. Thus, the
paths $\gamma_{w,\alpha_{i}}$ connect $w$ and $w^{\prime}=\sigma_{\alpha}\cdot
w$; since these paths coincide (up to reparametrization) with the trajectories
of the Toda system, which is gradient system with stable points equal to
$W(\mathfrak{g})$, we conclude from the definition of Bruhat order that there
exist trajectories of Toda system, connecting any two elements of
$W(\mathfrak{g})$, which differ by only one reflection.

Let now $w^{\prime}\in W(\mathfrak{g})$ be any element, equal to the product
of exactly two reflections. Let us prove that there exists a trajectory of the
Toda system, connecting the unit $e\in W(\mathfrak{g})$ and $w^{\prime}$. But
it follows from the properties of Toda system, listed in the end of previous
section, that the dimension of unstable manifold of $w^{\prime}$ is equal to
the dimension of the corresponding Bruhat cell, i.e. equals $2$. On the other
hand, there can be no trajectories, entering $w^{\prime}$ from the elements
whose length is greater or equal to $2$ (since these points do not verify the
condition $w\prec w^{\prime}$). Similarly, the dimension of the space, spanned
by the trajectories, connecting $w^{\prime}$ with elements of length $1$, can
not exceed $1$ (it follows from the first observation in this section). Thus,
this space should consist of the trajectories coming from $e$ (up to a measure
zero subset, spanned by the trajectories from ‘‘intermediate’’ points). It
means that the set of such trajectories is nonempty, i.e. the corresponding
Bruhat cell and the dual Bruhat cell intersect; it also follows that the
dimension of this intersection is $2$.

Next, if $w\prec w^{\prime}$ and $l(w^{\prime})-l(w)=2$, we choose
$\Lambda^{\prime}$ so that $w$ is minimal (this is done as before by passing
to the function $F_{\Lambda}(\Psi\cdot w)=F_{Ad_{w}(\Lambda)}(\Psi)$ for
suitable $w$; one can also say that this amounts to the change of the set of
positive roots, or to the action of Weyl group on the Weyl chambers
of$\mathfrak{h}$). In this new setting $w$ will play the role of unit in Weyl
group, and $w^{\prime}$ will correspond to an element of length 2. Then as
before, we conclude that there exist trajectories of $T^{\Lambda^{\prime}}$
from $w$ to $w^{\prime}$, and the space of these trajectories is
$2$-dimensional. Now by the remark we made in the end of the previous section
this surface is also invariant for $T^{\Lambda}$, thus, since $w$ is local
minimum and $w^{\prime}$ is maximum of $F_{\Lambda}$ on this surface (in fact
$F_{\Lambda^{\prime}}=w^{*}(F_{\Lambda})$ and the Riemannian structure used to
determine the Toda flow is (right) invariant) we conclude that there exists a
$2$-dimensional space of trajectories of $T^{\Lambda}$ from $w$ to
$w^{\prime}$, and hence the corresponding Bruhat and dual Bruhat cell
intersect along a $2$-dimensional variety.

Finally, we proceed by induction: assuming that we know the existence of
intersections of all cells with $l(w)-l(w^{\prime})\leq k$, we consider first
$w^{\prime}=e,l(w)=k+1$ and conclude (by dimension counting) that the space of
trajectories from $e$ to $w$ is nonempty and has dimension $k+1$; then, using
the shift we reduce the general case to this one.

Observe, that in addition to the main proposition we have also proved that to
stable points $w$ and $w^{\prime}$ of Toda system on a normal real form are
connected by a trajectory iff they are comparable in Bruhat order.

######  Remark 3.1.

It seems that the same intersection result can be obtained from the standard
reasoning, based on the combinatoric considerations and on the use of Bott-
Samelson varieties and their generalizations, similarly to the Deodhar’s
construction for algebraically-closed fields, see [10].

###  3.1 Non-split case and further questions

In non-split case we have the problem of identifying the real Bruhat cells
with the real part of complex cells. Thus the very first stage of our
inductive process fails. In fact, the Weyl group
$W(\mathfrak{g},\mathfrak{k})$ in this case coincides with the Weyl group of
the smaller root system, namely, of the so-called Satake projection, in which
$\mathfrak{a}$ plays the role of Cartan algebra. Besides this, some of the
root spaces $\mathfrak{g}_{\alpha}$ can be multidimensional.

On the other hand, most part of the statements from the previous section
remain true: the points $w\in W(\mathfrak{g},\mathfrak{k})$ are invariant
points of Toda system, their Bruhat cells and dual Bruhat cells (or their
shifted versions) are unstable and stable manifolds of the Toda system (see
[9]) and the paths $\gamma_{\alpha,w}$ are invariant with respect to Toda
flow.

Using these facts, one can make the following conjecture, concerning the
structure of trajectories of Toda flow on non-split real forms, and the
intersections of the Bruhat cells:

######  Conjecture 3.2.

There exist trajectories, connecting $w$ and $w^{\prime}$ in
$W(\mathfrak{g},\mathfrak{k})$ iff the corresponding elements are comparable
in Bruhat order, in which case the space of such trajectories coincides with
the intersection of Bruhat and dual Bruhat cell, and its dimension is equal to
the sum:

$\sum_{i=1}^{k}\dim\mathfrak{g}_{\alpha_{i}},$

where $w=\sigma_{\alpha_{i_{k}}}\cdot\dots\cdot\sigma_{\alpha_{i_{1}}}\cdot
w^{\prime}$. In particular, this sum does not depend on the choice of the
reflections $\sigma_{\alpha_{i}}$ in this formula.

Another interesting question is related to the fact, that in Deodhar’s work
([10]) one obtains further decomposition of the intersections of Bruhat cells
into smaller algebraic varieties. One can ask, if these varieties can also be
somehow characterized in terms of the Toda system.

#### Acknowledgments

The authors thank Sergey Loktev, Vladimir Rubtsov, Evgeniy Smirnov for useful
discussions. The work of Yu.B. Chernyakov was partly supported by the grant
RFBR-18-01-00461. The work of G.I Sharygin was supported by RNF grant
16-11-10069. The work of A.S. Sorin was supported by the DFG Grant LE
838/12-2. The work of D.V. Talalaev was supported by the RFBR grant No.
17-01-00366A.

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