§ INTRODUCTION

A precise measurement of the integrated luminosity is a key component
of the ATLAS physics programme at the CERN Large Hadron Collider (LHC),
in particular for cross-section
measurements where it is often one of the leading sources of uncertainty.
Searches for new physics phenomena beyond those predicted by the Standard Model
also often require accurate estimates of the luminosity to determine
background levels and sensitivity. This paper describes the measurement
of the luminosity of the proton–proton ($pp$)
collision data sample delivered to the
ATLAS detector at a centre-of-mass energy of  during Run 2 of
the LHC in the years 2015–2018.
The measurement builds on the experience and techniques developed for Run 1 and
documented extensively in Refs. <cit.>,
where uncertainties in the total integrated luminosities of
$\Delta{\cal L}/{\cal L}=1.8$% for  and 1.9% for  were achieved.

The luminosity measurement is based on an absolute calibration of the
primary luminosity-sensitive detectors in low-luminosity runs with
specially tailored LHC conditions using the van der Meer (vdM)
method <cit.>. A calibration transfer procedure was then used to
transport this calibration to the physics data-taking regime at high
luminosity. The vdM  calibration was performed in dedicated fills once per
year during Run 2, and relative comparisons of the luminosities measured by
different detectors were used to set limits on any possible change in the
calibration during the year. Finally, the integrated luminosity and
uncertainty for the whole Run 2 data-taking period was derived, taking into
account correlations between the uncertainties in each of the component years.

The structure of this paper mirrors the calibration process.
After a brief introduction
to the methodology and the Run 2 dataset in Section <ref>, the various
luminosity-sensitive detectors are described in Section <ref>,
the absolute vdM calibration in Section <ref>, the
calibration transfer procedure in Sections <ref> and <ref>,
the long-term stability studies in Section <ref> and the correlations
and combination in Section <ref>. The latter also contains
a summary (Table <ref>) of the contributing uncertainties in all
individual years and the combination. A dedicated luminosity calibration
for special datasets recorded for precision $W/Z$ physics with low numbers
of $pp$ interactions per bunch crossing is described in Section <ref>.
Conclusions are given in Section <ref>.

§ METHODOLOGY AND DATASETS

The instantaneous luminosity  produced by a single pair of colliding
bunches can be expressed as

\begin{equation}\label{e:lbungen}
\lbun=\frac{R}{\sigma}\ ,
\end{equation}

where $R$ is the rate of a particular process with cross-section $\sigma$.
In the case of inelastic $pp$ collisions
with cross-section , Eq. (<ref>) becomes

\begin{equation}\label{e:lbuninel}
\lbun=\frac{\mu f_\mathrm{r}}{\sigmainel}\ ,
\end{equation}

where the pileup parameter $\mu$ is the average number of inelastic interactions
per bunch crossing, and $f_\mathrm{r}$ is the LHC bunch revolution frequency
(11246 Hz for protons).
The total instantaneous luminosity is given by

\begin{equation}\nonumber
\linst=\sum_{b=1}^{\nbun} \lbun = \nbun\langle{\lbun}\rangle = \nbun \frac{\meanmu f_\mathrm{r}}{\sigmainel}\ ,
\end{equation}

where the sum runs over the  bunch pairs colliding at the interaction
point (IP), $\langle\lbun\rangle$ is the mean per-bunch luminosity and
is the pileup parameter averaged over all colliding bunch pairs.

The instantaneous luminosity can be measured by monitoring , the visible
interaction rate per bunch-crossing for a particular luminosity algorithm,
based on a chosen luminosity-sensitive detector.[The term `luminosity
algorithm' refers
to the measurement of a quantity proportional to the instantaneous luminosity,
e.g. a counting rate or electrical current, using a procedure or set of
selection criteria specific to a particular luminosity detector.]
The per-bunch luminosity can be written in analogy with Eqs. (<ref>)
and (<ref>) as

\begin{equation}\label{e:lbunvis}
\lbun=\frac{\muvis f_\mathrm{r}}{\sigmavis}\ ,
\end{equation}

where the algorithm-specific
visible cross-section is a calibration constant which represents the
absolute luminosity calibration of the given algorithm, and can be determined
via the vdM calibration method discussed in Section <ref>.
Once  is known, the pileup parameter $\mu$ can be determined
from $\mu=\muvis\sigmainel/\sigmavis$. Since  is not precisely
known, a reference value of $\sigmainel=80$ mb is used by convention
by all LHC experiments for
$pp$ collisions at . The values of $\mu$ and its bunch-averaged
counterpart  are important parameters for characterising the
performance of the LHC machine and detectors, but the luminosity calibration
based on  is independent of the chosen reference value of .

Provided the calibration constant is stable in time, and does not depend on the LHC conditions
(e.g. the number of bunches or value of ), the measurement of
at any moment is sufficient to determine the per-bunch
instantaneous luminosity at that time. In practice, no luminosity algorithm
is perfectly stable in time and independent of LHC conditions, and comparisons
between many different algorithms and detectors, each with their own strengths
and weaknesses, are essential to produce a precise luminosity measurement with
a robust uncertainty estimate.

In ATLAS, the rate  is measured over a finite time interval
called a luminosity block (LB), during which data-taking conditions (including
the instantaneous luminosity) are assumed to remain stable. The start and
end times of each LB are defined by the ATLAS central trigger processor, and
the normal duration during Run 2 data-taking was one minute, sometimes cut short
if a significant change of conditions such as a detector problem occurred.
The instantaneous luminosity for each LB was calculated from the sum of
per-bunch luminosities, each derived using Eq. (<ref>) and the
per-bunch values measured within the LB, modified by the calibration transfer
corrections discussed in Section <ref>. The integrated luminosity
was then obtained by multiplying  by the duration of the LB, and these
values were stored in the ATLAS conditions database <cit.>.
The data-taking is organised into a series of `runs' of the ATLAS data
acquisition system, each of which normally corresponds to an LHC fill,
including the periods outside stable beam collisions such as LHC injection,
acceleration and preparation for collisions.
ATLAS physics analyses are based on a `good runs list' (GRL), a list of runs
and LBs within them with stable beam collisions and
where all the ATLAS subdetectors were functioning correctly according to
a standard set of data-quality requirements <cit.>.
The integrated luminosity corresponding to the GRL can be  calculated as
the sum of the integrated luminosities of all LBs in the GRL, after correcting
each LB for readout dead-time. If the trigger used for a particular analysis
is prescaled, the effective integrated luminosity is reduced accordingly

The data-taking conditions evolved significantly during Run 2, with the LHC
peak instantaneous luminosity at the start of fills () increasing from
5 to $19\times 10^{33}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$
as the number of colliding bunch pairs  and average current per bunch
were increased during the course of each year. In addition,
progressively stronger focusing in the
ATLAS and CMS interaction regions (characterised by the $\beta^*$ parameter,
the value of the $\beta$ function at the interaction point
<cit.>) was used in each successive year,
leading to smaller transverse beam sizes and higher per-bunch
instantaneous luminosity.  Table <ref> shows
an overview of typical best parameters for each year, together with the
total delivered integrated luminosity. All this running took place with
long `trains' of bunches featuring 25 ns bunch spacing within the trains,
except for the second part of 2017, where a special filling pattern
with eight filled bunches separated by 25 ns followed by a four bunch-slot
gap (denoted `8b4e') was used. This bunch pattern mitigated the enhancement of
electron-cloud induced instabilities caused by a vacuum incident.
The luminosity was levelled by partial beam separation at
the beginning of such 8b4e LHC fills to give a maximum pileup parameter
of $\meanmu\approx 60$, whereas the maximum  achieved in standard
25 ns running was about 55 in 2018. These values are significantly larger
than the maximum  of around 40 achieved with 50 ns bunch spacing
in Run 1 <cit.>, and posed substantial challenges to the detectors.

Selected LHC parameters for $pp$ collisions at
in 2015–2018. The values shown are representative of the best
accelerator performance during normal physics operation. In 2017, the LHC
was run in two modes: standard 25 ns bunch train operation with long trains,
and `8b4e', denoting a pattern of eight bunches separated by 25 ns followed by a four bunch-slot gap. Values are given
for both configurations. The instantaneous luminosity was levelled by beam
separation to about $\lpeak=16\times 10^{33}\,\mathrm{cm^{-2}s^{-1}}$ for part of the 8b4e
period. The $0.1\,\ifb$ of physics data delivered during 2015 with 50 ns
bunch spacing is not included.
Parameter     2015     2016     2017     2018

Maximum number of colliding bunch pairs  ($n_b$)     2232     2208     2544/1909     2544

Bunch spacing [ns]     25     25     25/8b4e     25

Typical bunch population [$10^{11}$ protons]     1.1     1.1     1.1/1.2     1.1

$\beta^{*}$ [m]     0.8     0.4     0.3      0.3–0.25

Peak luminosity  [$10^{33}\,\mathrm{cm^{-2}s^{-1}}]$     5     13     16     19

Peak number of inelastic interactions/crossing ()     $\sim 16 \phantom{\sim\ }$     $\sim 41 \phantom{\sim\ }$     $\sim 45/60 \phantom{\sim}$     $\sim 55 \phantom{\sim\ }$

Luminosity-weighted mean inelastic interactions/crossing     13     25     38     36

Total delivered integrated luminosity []     4.0     39.0     50.6     63.8

§ LUMINOSITY DETECTORS AND ALGORITHMS

The ATLAS detector <cit.>
consists of an
inner tracking  detector surrounded by a thin superconducting solenoid
producing a 2 T axial magnetic field, electromagnetic and hadronic
calorimeters, and an external muon spectrometer incorporating three
large toroidal magnet assemblies.[ATLAS
uses a right-handed coordinate system with its origin at
the nominal interaction point in the centre of the detector, and the $z$ axis
along the beam line. The $x$ axis points from the nominal interaction point
to the centre of the LHC ring, and the $y$ axis points upwards. The two
sides of the detector are denoted A (corresponding to $z>0$) and C ($z<0$).
Pseudorapidity is defined in terms of the polar angle
$\theta$ as $\eta=-\ln\tan{\theta/2}$, and transverse momentum and energy
are defined relative to the beam line as $\pt=p\sin\theta$ and
$\et=E\sin\theta$. The azimuthal angle around the beam line is denoted by
The ATLAS luminosity measurement relies on multiple redundant luminosity
detectors and algorithms, which have complementary capabilities and
different systematic uncertainties. For LHC Run 2, the primary bunch-by-bunch
luminosity measurement was provided by the LUCID2 Cherenkov
detector <cit.> in the far forward region,
upgraded from its Run 1 configuration and referred to hereafter as
LUCID. This was complemented by bunch-by-bunch measurements from the
ATLAS beam conditions monitor (BCM) diamond detectors, and from offline
measurements of the
multiplicity of reconstructed charged particles in randomly selected
colliding-bunch crossings (track counting). The ATLAS calorimeters provided
bunch-integrated measurements (i.e. summed over all bunches)
based on quantities proportional to
instantaneous luminosity: liquid-argon (LAr) gap currents in the case of
the endcap electromagnetic (EMEC) and forward (FCal) calorimeters,
and photomultiplier currents from the scintillator-tile hadronic
calorimeter (TileCal).
All these measurements are discussed in more detail below.

§.§ LUCID Cherenkov detector

The LUCID detector contains 16 photomultiplier tubes (PMTs) in each forward arm
of the ATLAS detector (side A and side C), placed around the beam pipe in
different azimuthal $\phi$ positions at approximately
$z=\pm 17$ m from the interaction point and covering the pseudorapidity
range $5.561<|\eta|<5.641$. Cherenkov light is produced in the quartz windows
of the PMTs, which are coated with $^{207}$Bi radioactive sources that provide
a calibration signal. Regular calibration runs were performed between LHC fills,
allowing the  high voltage applied to the PMTs to be adjusted so as to keep
either the charge or amplitude of the calibration pulse constant over time.
The LUCID detector was read out with dedicated electronics which provided
luminosity counts for each of the 3564 nominal LHC bunch slots.
These counts were integrated
over the duration of each luminosity block, and over shorter time periods
of 1–2 seconds
to provide real-time feedback to the LHC machine for beam optimisation.

Several algorithms were used to convert the raw signals from the PMTs to
luminosity measurements, using a single PMT or combining the information
from several PMTs in various ways. The simplest algorithm uses a single
PMT, and counts an `event' if there is a signal in the PMT above a given
threshold (a `hit'), corresponding to one or more inelastic $pp$ interactions
detected in a given bunch crossing.
Assuming that the number of inelastic $pp$ interactions
in a bunch crossing follows a Poisson distribution, the probability for
such an event $P_\mathrm{evt}$ is given in terms of the single-PMT
visible interaction rate $\muvis=\epsilon\mu$ by

\begin{equation*}
P_\mathrm{evt}=\frac{N_\mathrm{evt}}{N_\mathrm{BC}}=1-\mathrm{e}^{-\muvis}\ ,
\end{equation*}

where $N_\mathrm{evt}$ is the number of events counted in the luminosity block
and $N_\mathrm{BC}$ is the number of bunch crossings sampled (equal for a single
colliding bunch pair to
$f_\mathrm{r}\Delta t$ where $\Delta t$ is the duration of the luminosity block).
The value is then given by $\muvis=-\ln(1-P_\mathrm{evt})$, from which the
instantaneous luminosity can be calculated via Eq. (<ref>) once
is known.
Several PMTs can be combined in an `EventOR' (hereafter abbreviated to EvtOR)
algorithm by counting an event if
any  of a group of PMTs registers a hit in a given bunch crossing.
Such an algorithm has a larger efficiency and  than a single-PMT
algorithm, giving a smaller statistical uncertainty at low bunch luminosity.
However, at moderate single-bunch luminosity ($\mu>20$–30), it already suffers
from `zero starvation', when for a given colliding-bunch pair,
all bunch crossings in the LB considered contain at least one hit. In this
situation, there are no bunch crossings left without an event, making
it impossible to determine .
LUCID EvtOR algorithms were therefore not used for standard high-pileup
physics running during Run 2. The `HitOR' algorithm provides an alternative
method for combining PMTs at high luminosity. For $N_\mathrm{PMT}$ PMTs, the
average probability $P_\mathrm{hit}$ to have a hit in any given PMT
during the $N_\mathrm{BC}$ bunch crossings of one luminosity block is inferred
from the total number of hits summed over all PMTs $N_\mathrm{hit}$ by

\begin{equation}\label{e:phit}
\phit = \frac{N_\mathrm{hit}}{N_\mathrm{BC}N_\mathrm{PMT}} = 1-\mathrm{e}^{-\muvis} ,
\end{equation}

which leads for the HitOR algorithm to $\muvis=-\ln(1-\phit)$.
Per-bunch LB-averaged event and hit counts from a variety of PMT combinations
were accumulated online in the LUCID readout electronics.

Both the configuration of LUCID and the LHC running conditions evolved
significantly over the course of Run 2, and different algorithms
gave the best LUCID offline luminosity measurement in each data-taking year.
These algorithms are  summarised in Table <ref>, together with
the corresponding visible cross-sections derived from the vdM calibration,
the peak  value and the corresponding fraction of bunch crossings
which do not have a PMT hit. The latter fell below 1% at the highest
instantaneous luminosities achieved in 2017 and 2018.
Where possible, HitOR algorithms
were used, as the averaging over multiple PMTs reduces systematic uncertainties
due to drifts in the calibration of individual PMTs, and also evens out
the asymmetric response of PMTs in different $\phi$ locations due to the
LHC beam crossing angle used in bunch train running. The BiHitOR algorithm,
which combines four bismuth-calibrated PMTs on the A-side of LUCID with four
more on the C-side using the HitOR method,
was used in both 2016 and 2017. Some PMTs were changed
during each winter shutdown, so the sets of PMTs used in the two years
were not exactly the same. In 2015, nearly all PMTs suffered from long-term
timing drifts
due to adjustments of the high-voltage settings, apart from one PMT on
the C-side (C9), and the most stable luminosity measurement (determined from
comparisons with independent measurements from other detectors) was obtained
from the single-PMT algorithm applied to PMT C9.
In 2015 and 2016, the bismuth calibration
signals were used to adjust the PMT gains so as to keep the mean charge
recorded by the PMT constant over the year. However, the efficiencies of the
hit-counting algorithms depend primarily on the pulse amplitude. Additional
offline corrections were therefore applied to the LUCID data in these years to
for this imperfect calibration. In 2017 and 2018, the gain adjustments were
made using the amplitudes derived from the calibration signals, and so these
corrections were not needed. In 2018, a significant number of PMTs
stopped working during the course of the data-taking year, and a single PMT
on the C-side (C12) was used for the final offline measurement, as
it showed good stability throughout the year and gave results similar to those
of a more complicated offline HitOR-type combination of the remaining seven
working PMTs. In all years, other LUCID algorithms were also available, and
were studied as part of the vdM and stability analyses. These algorithms include
Bi2HitOR, an alternative to BiHitOR using an independent set of
eight PMTs with four on each side of the detector, single-sided HitOR and EvtOR
algorithms, EvtAND algorithms requiring a coincidence of hits on both sides,
and single-PMT algorithms based on individual PMTs.

LUCID algorithms used for the baseline luminosity
determination in each year of Run 2 data-taking, together with the visible
cross-sections  determined from the absolute vdM calibration
(see Section <ref>),
the peak  value taken from Table <ref>, and the fraction of
bunch crossings $f_\mathrm{no-hit}$ which do not have a PMT hit
at this  value. For the HitOR algorithms, this fraction
represents an average over all the contributing PMTs.
Year     Algorithm      [mb]     Peak      $f_\mathrm{no-hit}\,[\%]$

2015     PMT C9      6.540     16     27.0

2016     BiHitOR     6.525     41     3.5

2017     BiHitOR     6.706     60     0.6

2018     PMT C12     6.860     55     0.9

§.§ Beam conditions monitor

The BCM detector consists of four $8\times 8\,\mathrm{mm}^2$ diamond sensors
arranged around the beam pipe in a cross pattern at $z=\pm 1.84$ m on each
side of the ATLAS interaction point <cit.>.
The BCM mainly provides beam conditions
information and beam abort functionality to protect the ATLAS inner detector,
but it also gives a bunch-by-bunch luminosity signal with
sub-nanosecond timing resolution. Various luminosity algorithms are available,
combining hits from individual sensors in different ways with EvtOR and EvtAND
algorithms in the same way as for LUCID. The BCM provided the primary ATLAS
luminosity measurement during most of Run 1 <cit.>
but performed much less well in the physics data-taking conditions of Run 2,
with 25 ns bunch spacing and higher pileup, due to a combination of radiation
damage, charge-pumping effects <cit.> and
biases along bunch trains. Its use in the Run 2 analysis was therefore
limited to consistency checks during some vdM scan periods with isolated bunches
and low instantaneous luminosity, and luminosity measurement in
heavy-ion collisions.

§.§ Track-counting algorithms

The track-counting luminosity measurement determines the per-bunch visible
rate  from the mean number of reconstructed tracks per bunch crossing
averaged over a luminosity block. The measurement was derived from
randomly sampled colliding-bunch crossings, where only the data from the
silicon tracking detectors (i.e. the SCT
and pixel detectors, including a new innermost layer, the `insertable B-layer' (IBL) <cit.> added before Run 2)
were read out, typically at 200 Hz during normal physics data-taking and at
much higher rates during vdM scans and other special runs. These events
were saved in a dedicated event stream, which was then reconstructed
offline using special track reconstruction settings, optimised for luminosity
monitoring. All reconstructed tracks
used in the track-counting luminosity measurement were required to
satisfy the TightPrimary requirements of Ref. <cit.>,
and to have transverse momentum $\pt>0.9$ GeV and a track impact parameter
significance of
$|d_0|/\sigma_{d_0}<7$, where $d_0$ is the impact parameter of the
track with respect to the beamline in the transverse plane,
and $\sigma_{d_0}$ the uncertainty
in the measured $d_0$, including the transverse spread of the luminous region.
Several different track selection working points,
applying additional criteria on top of these basic requirements, were
used, as summarised in Table <ref>.
The baseline selection A uses tracks only in the
barrel region of the inner detector ($|\eta|<1.0$), requires
at least nine silicon hits  (counting both pixel and SCT hits), and
requires at most one pixel `hole' (), i.e. a missing pixel hit
where one is expected, taking into account known dead modules.
Selection B extends the acceptance into the endcap
tracking region, with tighter hit requirements for $|\eta|>1.65$, and
does not allow any pixel holes. Selection C is based on
selection A, with a tighter requirement of at least ten silicon hits.
Selection A was used as the baseline track-counting luminosity measurement,
and the other selections were used to study systematic uncertainties.

Track-selection criteria for the different working
points used in the track-counting luminosity measurement, applied in addition
to the basic TightPrimary selection of Ref. <cit.>.
Selection A was used for the baseline track-counting luminosity measurement.
Criterion     Selection A     Selection B     Selection C

[GeV]     $>0.9$     $>0.9$     $>0.9$

$|\eta|$     $<1.0$     $<2.5$     $<1.0$

$\geq 9$     $\geq 9$ if $|\eta|<1.65$     $\geq 10$

else $\geq 11$

$\leq 1$     $=0$     $\leq 1$

$|d_0|/\sigma_{d_0}$     $<7$     $<7$     $<7$

The statistical precision of the track-counting luminosity measurement is
limited  at low luminosity, making it impractical to calibrate it directly
using vdM scans. Instead, the working points were calibrated to agree with LUCID
luminosity measurements in the same LHC fills as the vdM scans, utilising
periods with stable, almost constant
luminosity where the beams were colliding head-on in ATLAS, typically while
vdM scans were being performed at the CMS interaction point.
The average number of selected tracks
per  inelastic $pp$ collision was about 1.7 for selections A and C,
and about 3.7 for selection B, which has a larger acceptance in $|\eta|$.

§.§ Calorimeter-based algorithms

The electromagnetic endcap calorimeters  are sampling calorimeters
with liquid argon as the active medium and lead/stainless-steel absorbers,
covering the region $1.5<|\eta|<3.2$ on each side of the ATLAS detector.
Ionisation electrons produced by charged particles crossing the LAr-filled
gaps between the absorbers draw a current through the high-voltage (HV) power
supplies that is proportional to the instantaneous luminosity, after
subtracting the electronics pedestal determined from the period without
collisions at the start of each LHC fill. A subset of the EMEC HV lines
are used to produce a luminosity measurement, independently from the
A- and C-side EMEC detectors. A similar measurement is available from the
forward calorimeters that cover the region $3.2<|\eta|<4.9$ with longitudinal
segmentation into three modules. The first module has copper absorbers,
optimised for the detection of electromagnetic showers, and the HV currents from
the 16 azimuthal sectors are combined to produce a luminosity measurement,
again separately from the A- and C-side calorimeters. Since both the
EMEC and FCal luminosity measurements rely on `slow' ionisation currents,
they cannot resolve individual bunches and therefore give bunch-integrated
which are read out every few seconds and then averaged per luminosity block.
They have insufficient sensitivity at low luminosity to be calibrated during
vdM scans, so they were cross-calibrated to track-counting measurements in
high-luminosity physics fills as discussed in Section <ref>.

The event-by-event `energy flow' through the cells of the LAr calorimeters,
as read out by the standard LAr pulse-shaping electronics, provides another
potential luminosity measurement. However, the long LAr drift time and bipolar
pulse shaping <cit.> wash out any usable signal during running
with bunch trains. Only the EMEC and FCal cells have short enough drift
times to yield useful measurements in runs with isolated bunches
separated by at least 500 ns. Under these conditions, the total energy
deposited in a group of LAr cells and averaged over a luminosity block is
proportional to the instantaneous luminosity (after pedestal subtraction). A
dedicated data stream reading out only the LAr calorimeter data in
randomly sampled bunch crossings was used in certain runs with isolated bunches
to derive a LAr energy-flow luminosity measurement. This was exploited
to constrain potential non-linearity in the track-counting luminosity
measurement as discussed in Section <ref>.

The TileCal hadronic calorimeter uses steel absorber plates interleaved with
plastic scintillators as the active medium, read out by wavelength-shifting
fibres connected to PMTs <cit.>.
It is divided into a central barrel section covering
$|\eta|<1.0$, and an extended barrel on each side of the detector (A and C)
covering $0.8<|\eta|<1.7$. Both the barrel and extended barrels are segmented
azimuthally into 64 sectors, and longitudinally into three sections.
The current drawn by each PMT (after pedestal correction) is proportional to
the total number of particles traversing the corresponding TileCal cell, and
hence to the instantaneous luminosity. In principle, all TileCal cells can be
used for luminosity measurement, but the D-cells, located at the largest radius
in the last longitudinal sampling, suffer least from variations in response
over time due to radiation damage. The D6 cells, at the highest pseudorapidity
in the extended barrels on the A- and C-sides, were used for the primary
TileCal luminosity
measurements in this analysis, with other D-cells at lower pseudorapidity
being used for systematic comparisons. Changes in the gains of the PMTs with
time were monitored between physics runs
using a laser calibration system which injects pulses
directly into the PMTs, and the response of the scintillators was also
monitored once or twice per year using $^{137}$Cs radioactive sources
circulating through the calorimeter cells during shutdown periods. The TileCal
D-cells have insufficient sensitivity to be calibrated during vdM scans, so
they were cross-calibrated to track-counting in the same way as for the EMEC
and FCal measurements.

The TileCal system also includes the E1 to E4 scintillators, installed
in the gaps between the barrel and endcap calorimeter assemblies, and designed
primarily to measure energy loss in this region. Being much more exposed to
particles from the interaction point than the rest of the TileCal scintillators
(which are shielded by the electromagnetic calorimeters)
the E-cells (in particular E3, and E4 which is closest to the beamline) are
sensitive enough to make precise luminosity measurements during vdM fills.
They play a vital role in constraining
the calibration transfer uncertainties, as discussed in Section <ref>.
However, their response changes rapidly over time due to radiation damage,
so they cannot be used for long-term stability studies. The high currents
drawn by the E-cell PMTs also cause the PMT gains to increase slightly
at high luminosity, resulting in an overestimation of the actual luminosity.
This effect was monitored during collision data-taking by firing the
TileCal calibration laser periodically during the LHC abort gap—a time window
during each revolution of the LHC beams, in which no bunches pass through the ATLAS interaction point, and which is left free for the duration of the
rise time of the LHC beam-dump kicker magnets.
By comparing, in 2018 data, the response of the E-cell PMTs  to the laser pulse
with that of the D6-cell PMTs (the gain of which remains stable thanks to the
much lower currents), a correction to the E-cell PMT gain was derived, reducing
the reported luminosity in high-luminosity running by up to 1% for the E4
cells, and somewhat less for E3.
This correction is applied to all the TileCal E-cell data shown in this paper.

§ ABSOLUTE LUMINOSITY CALIBRATION IN VDM SCANS

The absolute luminosity calibration of LUCID and BCM, corresponding
to the determination of the visible cross-section  for each of
the LUCID and BCM algorithms, was derived using dedicated vdM scan sessions
during special LHC fills in each data-taking year. The calibration
methodology and main sources of uncertainty are similar to those in
Run 1 <cit.>,
but with significant refinements in the light of Run 2
experience and additional studies. The improvements especially concern
the scan-curve fitting, the treatment of beam–beam and emittance
growth effects, and the
potential effects of non-linearity from magnetic hysteresis in the LHC
steering corrector magnets used to move the beams in vdM scans.
These aspects are given particular attention below.
The uncertainties related to each aspect of the vdM calibration are discussed
in the relevant subsections, and listed for each data-taking year
in Table <ref> in Section <ref>.

§.§ vdM formalism

The instantaneous luminosity for a single colliding bunch pair, , is given
in terms of LHC beam parameters by

\begin{equation}\label{e:vdm}
\lbun = \frac{f_\mathrm{r} n_1 n_2}{2\pi \capsigx \capsigy}\ ,
\end{equation}

where $f_\mathrm{r}$ is the LHC revolution frequency,
$n_1$ and $n_2$ are the numbers of protons in the beam-1 and beam-2 colliding
bunches, and  and  are the convolved beam sizes
in the horizontal and vertical (transverse) planes <cit.>.
In the vdM method, the uncalibrated instantaneous luminosity or counting rate
$R(\Delta x)$ is measured as a function of the nominal separation
$\Delta x$ between the two beams in the horizontal plane, given by

\begin{equation}\nonumber
\Delta x=\xpos{1}-\xpos{2}\ ,
\end{equation}

where i is the displacement of beam $i$ from its initial position
(corresponding to $\xpos{i}=0$) with approximately head-on collisions
(i.e. no
nominal transverse separation). These initial positions were established by
optimising the luminosity as a function of separation in each plane
before starting the first vdM scan in a session. The separation was then
varied in steps by displacing the two beams in opposite directions
($\xpos{2}=-\xpos{1}$), starting with the maximum negative separation,
moving through the peak and onward to maximum positive separation,
and measuring the instantaneous luminosity $R(\Delta x)$ at each step.
The procedure was then repeated in the vertical plane (varying $\Delta y$ and
keeping $\Delta x=0$) to measure $R(\Delta y)$,
completing an $x$–$y$ scan pair.
The quantity  is then given by

\begin{equation}\label{e:capsig}
\capsigx = \frac{1}{\sqrt{2\pi}} \frac{\int R(\Delta x)\, \mathrm{d}\Delta x}{R(\dxmax)}\ ,
\end{equation}

i.e. the ratio of the integral over the horizontal scan to the
instantaneous luminosity $R(\dxmax)$ on the peak of the scan
at $\dxmax\approx 0$, and similarly for  with a vertical

If the form of $R(\Delta x)$ is Gaussian,  is equal to the
standard deviation of the distribution, but the method is valid for any
functional form of $R(\Delta x)$.  However, the formulation does assume that
the particle densities in each bunch can be factorised into independent
functions of $x$ and $y$. The effect of violations of this assumption
(i.e. of non-factorisation) is quantified in Section <ref> below.
Since the normalisation of $R(\Delta x)$
cancels out in Eq. (<ref>), any quantity proportional to luminosity
can be used to determine the scan curve. The calibration of a given algorithm
(i.e. its  value) can then be determined by combining
Eqs. (<ref>) and (<ref>) to give

\begin{equation}\label{e:sigcal}
\sigmavis = \muvismax \frac{2\pi \capsigx\capsigy}{n_1 n_2}\ ,
\end{equation}

where $\muvismax$ is the visible interaction rate per bunch crossing
at the peak of the scan curve. In practice, the vdM scan curve was analysed
in terms of the specific visible interaction rate , the visible
interaction rate  divided  by the bunch population product $n_1n_2$
measured at each scan step, as discussed in Section <ref>.
This normalisation accounts for the small
decreases in bunch populations during the course of each scan. The specific
interaction rate in head-on collisions  was determined from the
average of the rates fitted at the peaks of the $x$ and $y$ scan curves
x and y, so Eq. (<ref>) becomes

\begin{equation}\label{e:sigcalav}
\sigmavis=\frac{1}{2}\left( \mubvismaxi{x}+\mubvismaxi{y} \right)\cdot 2\pi\capsigx\capsigy\ .
\end{equation}

A single pair of $x$–$y$ vdM scans
suffices to determine  for each algorithm active during the
scan. Since the quantities entering Eq. (<ref>) may be different
for each colliding bunch pair, it is essential to perform a bunch-by-bunch
to determine , limiting the vdM absolute calibration
in ATLAS to the LUCID and BCM luminosity algorithms. Track-counting cannot
be used since only a limited rate of bunch crossings
can be read out for offline analysis,[At , track-counting
algorithms were calibrated in vdM scans <cit.>, but trigger-rate
and dataflow limitations restricted this calibration to a small subset
of the colliding bunch pairs. A more precise calibration can be achieved
by cross-calibrating to LUCID algorithms as discussed in
Section <ref>.]
and the calorimeter algorithms only measure bunch-integrated luminosity.

§.§ vdM scan datasets

In order to minimise both instrumental and accelerator-related uncertainties
<cit.>,  vdM scans were not performed in standard physics
conditions, but once per year in dedicated low-luminosity vdM scan fills using
a special LHC configuration. Filling schemes with 44–140 isolated
bunches in each beam were
used, so as to avoid the parasitic encounters between incoming and outgoing
bunches that occur in normal bunch train running. This allowed the beam
crossing angle to be set to zero, reducing orbit drift and beam–beam-related
uncertainties. The parameter $\beta^*$ was set to 19.2 m rather than the
0.25–0.8 m used during normal physics running, and the injected
beam emittance was increased to 3–4 $\mu$m-rad. These changes
increased the RMS transverse size of the luminous region to about $60\,\mu$m,
thus reducing vertex-resolution uncertainties in evaluating the
non-factorisation corrections discussed below.
The longitudinal size of the luminous region was typically around
45 mm, corresponding to an RMS bunch length of 64 mm.
Special care was taken in the LHC injector chain to produce beams with
Gaussian-like transverse profiles <cit.>, a procedure that also
minimises non-factorisation effects in the scans.
Finally, bunch currents were reduced to $\sim 0.8\times 10^{11}$ protons/bunch,
minimising bunch-current normalisation uncertainties. When combined with
the enlarged emittance, this also reduces beam–beam biases.
These configurations typically resulted in
$\meanmu\approx 0.5$ at the peak of the scan curves, whilst maintaining
measurable rates in the tails of the scans at up to  $6\signom$
separation, where $\signom\approx 100\,\mu$m is the nominal single-beam size
at the interaction point.

The vdM scan datasets are listed in Table <ref>. Several pairs of
on-axis $x$–$y$ scans were performed in each session, spaced
through one or more fills in order to study reproducibility. One or two pairs of
off-axis scans, where the beam was scanned in $x$ with an offset in $y$ of
e.g. $300\,\mu$m (and vice versa) were also included,
to study the beam profiles in the tails
in order to constrain non-factorisation effects. A diagonal scan varying
$x$ and $y$ simultaneously was also performed in 2016.
A typical scan pair took $2\times 20$ minutes for $x$ and $y$, scanning
the beam separation between $\pm 6\signom$ in 25 steps of $0.5\signom$.
Length scale
calibration (LSC) scans were also performed in the vdM or nearby fills,
as described in Section <ref>. A vdM fill lasted up to one day,
with alternating groups of scans in ATLAS and CMS, and the beams colliding
head-on in the experiment which was not performing scans.

Summary of the  $pp$ vdM scan fills used in the
ATLAS Run 2 vdM analysis, showing the dates, LHC fill numbers, numbers of
bunches in each beam (), numbers of bunch pairs colliding in ATLAS
(), and the individual $x$–$y$ scan pairs performed in each fill
(S1, S2, etc., including off-axis scans denoted `Sn-off', a
diagonal $xy$ scan in 2016 denoted `S5-diag' and length scale
calibration scans denoted LSC). All fills
used the dedicated vdM optics with $\beta^*=19.2$ m, with zero beam crossing
angle and only isolated bunches.
Date     Fill             Scans

24/8/2015     4266     44     30     S1, S2, S3-off

25/8/2015     4269     51     8     S4, LSC

17/5/2016     4937     55     11     LSC

18/5/2016     4945     52     32     S1,S2, S3-off, S4, S5-diag

27/5/2016     4954     52     32     S6

28/7/2017     6016     52     32     S1, S2, S3-off, S4, S5, LSC

29/6/2018     6864     70     58     LSC

30/6/2018     6868     140     124     S1, S2, S3-off, S4, S5-off, S6

§.§ Scan curve fitting

Typical vdM scan curves from the 2017 and 2018 datasets using the LUCID BiHitOR
and C12 algorithms are shown in Figure <ref>.
The  distributions
were fitted using an analytic function (e.g. a Gaussian function multiplied
by a polynomial), after subtraction of backgrounds. For the LUCID algorithms,
the background is dominated by noise from the $^{207}$Bi calibration
source, which was estimated by subtracting the counting rate measured
in the preceding bunch slot. The latter was always empty in the LHC filling
patterns used for vdM scans, which only include isolated bunches separated
by at least 20 unfilled bunch slots.
This procedure also subtracts the smaller
background due to `afterglow' from photons produced in the decay of nuclei
produced in hadronic showers initiated by the $pp$ collisions. This background
scales with
the instantaneous luminosity at each scan point, and decays slowly over the
bunch slots following each filled bunch slot. A further small background
comes from the interaction of the protons in each beam with residual gas
molecules in the beam pipe, and scales with the bunch population
$n_1$ or $n_2$. It was estimated using the counting rate in the
bunch slots corresponding to `unpaired' bunches,
where a bunch from one beam passes through the ATLAS interaction point
without meeting a bunch travelling in the opposite direction.

Visible interaction rate  per unit bunch
population product $n_1n_2$ (i.e. )
vs. beam separation in (a) the horizontal plane
for bunch slot (BCID) 1112 in scan 1 of 2017, using the LUCID BiHitOR algorithm,
and (b) the vertical plane for BCID 34 in scan 2 of 2018, using the
LUCID C12 PMT algorithm. The measured rate is shown by the pink squares,
the background-subtracted rate by the red circles, the estimated background
from noise and afterglow by the blue upward-pointing triangles and that from
beam–gas interactions by the green downward-pointing triangles.
The error bars show the statistical uncertainties, which are in some cases
smaller than the symbol sizes.
The fits to a Gaussian function multiplied by a fourth-order polynomial (GP4)
plus an additional Gaussian function in the central region and a constant term
(see text) are shown by the dashed red lines.

Since the visible interaction rate is only sampled at a limited number of scan
points, the choice of fit function is crucial in order to ensure unbiased
estimates of  and . Fitting the scan curves to a Gaussian
function multiplied by a fourth-order polynomial (referred to as GP4)
gives a good general description
of the shape, but was found to systematically overestimate the peak of the
scan curves by about 0.5%, biasing . Better results were achieved
by adding a second Gaussian function to the GP4 function for the five
central scan points, flattening the scan curve at the peak.[The second
Gaussian function introduces discontinuities into the fitting function just
outside the five central points, but the results were found not to depend on
whether the discontinuities were placed close to the central points or close to
the next two scan points at larger $|\Delta x|$.] This GP4+G function
was used for the baseline  results. The statistical uncertainties
from the fit are smaller than 0.1% in all years, and are listed as
`Statistical uncertainty' in Table <ref>.
A Gaussian function multiplied by a
sixth-order polynomial (GP6) was found to give a similar goodness-of-fit
to GP4+G, and the difference between the  obtained from these two
functions was used to define the `Fit model' uncertainty shown for each
data-taking year in Table <ref>.
In all cases, a constant term was also included in the fit function, and the
integrals to determine  (Eq. (<ref>)) were truncated at the
limits of the scan range. The constant term improves the fit stability and the
description of the tails.

To estimate the uncertainty due to the background subtraction procedure,
alternative fits were performed to the data without subtracting
the estimates of the noise and beam–gas components, instead using the constant
term to account for their contributions, and
implicitly assuming that the backgrounds do not depend on beam separation.
In this case, the constant term
was not included in the evaluation of  and .
The resulting difference in  was used to define the `Background
subtraction' uncertainty in Table <ref>.

The vdM scan curve fits were performed for all available LUCID and BCM
algorithms, and the results are compared in Section <ref>. However,
most of the systematic uncertainty studies detailed below were only performed
with a single `reference' algorithm for each year of data-taking.
For 2016–18, this is the algorithm
listed in Table <ref> and used for the baseline luminosity
determination; however for 2015, the LUCID BiEvtORA algorithm, using
four PMTs on the A-side of LUCID in an EvtOR combination, was used as the
reference algorithm instead.

§.§ Bunch population measurement

The determination of the bunch populations $n_1$ and $n_2$
was based on the LHC beam instrumentation, using
the methods discussed in detail in Ref. <cit.>.
The total intensity in each beam was measured by the LHC DC current
transformers (DCCT), which have an absolute precision of better than 0.1% but
lack the ability to resolve individual bunches. The per-bunch intensities
were determined from the fast beam-current transformers (FBCT), which
can resolve the current in each of the 3564 nominal 25 ns bunch slots in
each beam. The FBCT measurements were  normalised to the DCCT measurement at
each vdM scan step, and
supplemented by measurements from the ATLAS beam-pickup timing
system (BPTX), which can also resolve the relative populations in each bunch.

Both the FBCT and BPTX systems suffer from non-linear behaviour, which
can be parameterised as a constant offset between the measured and
true beam current, assuming the measurement is approximately linear near
the average bunch currents $\bar{n}_1$ and $\bar{n}_2$ for the two beams.
For the BPTX measurements, these offsets were determined
for each $x$–$y$ scan pair from the data by minimising the differences between
the corrected  values measured for all bunches, defining a
$\chi^2$ by

\begin{equation}\nonumber
\chi^2(\sigbarvis,b_1,b_2)=\sum_i \left(\frac{S(\sigbarvis,b_1,b_2,n^i_1,n^i_2)-\siguncvisi}{\Delta\siguncvisi}\right)^2\ ,
\end{equation}

where $\siguncvisi$ is the uncorrected measured visible cross-section for
colliding bunch pair $i$ and $\Delta\siguncvisi$ its statistical error,
and $S$ represents the modified visible cross-section
which would be measured for bunch pair $i$, given a true average visible cross-section $\sigbarvis$ and bunch current offsets $b_1$ and $b_2$ for the two beams.
The quantity $S$ is given by

\begin{equation}\nonumber
S(\sigbarvis,b_1,b_2,n^i_1,n^i_2)=\sigbarvis\cdot\left(\frac{n^i_1+b_1(n^i_1-\bar{n}_1)}{n^i_1}\right) \left(\frac{n^i_2+b_2(n^i_2-\bar{n}_2)}{n^i_2}\right)\ ,
\end{equation}

where the two terms in brackets represent the bunch-dependent biases in
the measured  values caused by the non-linearity in the BPTX bunch
current measurements. The values of $b_1$ and $b_2$ were determined separately
for each scan in each year by minimising the $\chi^2$ as a function of
$b_1$, $b_2$ and $\sigbarvis$ <cit.>, and have typical sizes of a
few percent.
In all vdM datasets, this correction reduces the apparent dependence of the
measured for each bunch pair on the BPTX bunch-population product

The FBCT uses independent electronics channels with separate scales and
offsets for the even- and odd-numbered bunch slots, making it difficult to apply
the above procedure to the FBCT measurements. Instead, the uncorrected FBCT
measurements in each scan were fitted to a linear function of the
offset-fit-corrected BPTX measurements, deriving parameters $p_0^j$
and $p_1^j$, where $j=1$ for odd and 2 for even bunch pairs.
These parameters, derived from separate fits to the odd and even
bunches within each scan, were used to determine corrected FBCT populations
$n_1^{i,j,\mathrm{FBCT-corr}}$ from the uncorrected measurements
$n_1^{i,j,\mathrm{FBCT-uncorr}}$ via

\begin{equation}\nonumber
n_1^{i,j,\mathrm{FBCT-corr}} = (n_1^{i,j,\mathrm{FBCT-uncorr}}-p_0^j)/p_1^j \ ,
\end{equation}

and similarly for $n_2^{i,j,\mathrm{FBCT-corr}}$.
This correction procedure reduced the $n_1n_2$ dependence of the values measured using the FBCT, with residual variations contributing
to the bunch-by-bunch consistency uncertainty discussed in
Section <ref> below.
The FBCT bunch population measurements (corrected
using the BPTX offset fit) were used for the baseline  results,
with the alternative BPTX-only results used to define the `FBCT bunch-by-bunch
fractions' uncertainty in Table <ref>, which is below 0.1% for all years.
The changes in  resulting from these corrections amount to about
+0.4% in 2015 and less than $\pm 0.05$% in all other years.

The DCCT measurement used to normalise the BPTX and FBCT results summed over
all filled bunches is also sensitive to additional contributions to the
total circulating beam currents from
ghost charge and satellite bunches. Ghost charge refers to circulating protons
present in nominally unfilled bunch slots, which are included in the
DCCT reading but are typically below the FBCT or BPTX threshold.
Their contribution was measured by comparing the rate of beam–gas vertices
reconstructed by the LHCb experiment in nominally empty bunch slots with the
rate in those bunch slots containing an unpaired bunch <cit.>.

Satellite bunches are composed of protons present in
a nominally filled bunch slot, but which are captured in a radio-frequency (RF)
bucket at least one RF period (2.5 ns) away from the nominally filled bucket.
Their current is included in the FBCT reading, but because collisions between
satellite bunches in each beam
make a negligible contribution to the nominal luminosity signal
at the interaction point, their contributions to the FBCT measurements
must be estimated and subtracted. Satellite populations were measured
using the LHC longitudinal density monitor <cit.>.
At zero crossing angle, longitudinally displaced collisions between a
satellite bunch and the opposing
in-time nominal bunch can contribute to the measured luminosity signal in the
corresponding bunch slot. The magnitude of this additional contribution
is proportional to the sum of the fractional satellite populations in the two
beams, and also depends on the emittance of the satellite bunch, on the
number of RF buckets by which it is offset from the nominal bunch, and
on the longitudinal location and time acceptance of the luminosity detector
considered.  The fractional satellite contribution
summed over both beams is at most 0.02%, 0.03%, 0.02% and 0.09%
in each of the datasets from 2015 to 2018, and these values were conservatively
taken as uncertainties in  due to the additional luminosity produced
by nominal–satellite collisions.

The combined effects of ghost and satellite charges on the bunch-population
measurement lead to positive corrections
to  of 0.1–0.4%, depending on the vdM fill in question, with
uncertainties that are an order of magnitude smaller. The corresponding
uncertainty in Table <ref> also includes the effect of nominal–satellite

A further uncertainty of 0.2% arises from the absolute calibration of the
DCCT current measurements, derived from a precision current source. The
uncertainty takes into account residual non-linearities, long-term stability
and dependence on beam conditions, and was derived using the procedures
described in Ref. <cit.>.

§.§ Orbit drift corrections

Gradual orbit drifts of up to $O(10\,\mu\mathrm{m})$ in the positions of
one or both beams have
been observed during the course of a single vdM scan. These drifts change the
actual beam separation from the nominal one produced by the LHC steering
corrector magnets, and were monitored using two independent beam position
monitor (BPM) systems. The DOROS BPM system[Named for the  DOROS
(Diode ORbit and Oscillation System) front-end electronics, which can
separate the close-in-time signals from the bunches circulating in opposite
directions in the same beam pipe close to the IP.]
uses pickups located at the inner
(i.e. closest to the IP) ends of the LHC final-focus quadrupole triplet
at $z=\pm 21.7$ m on each side of the IP, allowing the beam position at the IP
to be determined by averaging the measurements from each side. Since these
BPMs are located closer to ATLAS than the steering correctors used to move the
beams during vdM scans, they also see the beam displacements during scans.
The arc BPMs are located at intervals throughout the LHC magnetic arcs, and
the measurements from those close to ATLAS can be extrapolated to give the
beam position at the IP. The arc BPMs do not see the displacement during scans,
apart from any residual effects leaking into the LHC arcs due to non-closure
of the orbit bumps around the IP. For both BPM systems, the beam positions in
both planes were measured at the zero nominal beam separation points
immediately before and after each single scan ($x$ or $y$), and any changes
were interpolated linearly as a function of time within the scan
to give corrections $\delta^{x,y}_1(t)$ and $\delta^{x,y}_{2}(t)$ for beams 1
and 2.

The dominant effect on the vdM analysis comes from in-plane drifts,
defined as horizontal
(vertical) drifts during a horizontal (vertical) scan. They distort
the beam-separation scale in Figure <ref>, changing the
values of  obtained from Eq. (<ref>). These drifts were
taken into account by correcting the separation at each scan step to include
the  interpolated orbit drifts:

\begin{equation}\nonumber
\Delta x^\mathrm{corr}=\Delta x^\mathrm{nominal}+\delta^x_1-\delta^x_2\ ,
\end{equation}

and similarly for $y$, before fitting the scan curves.

A subdominant effect is caused by out-of-plane drifts between the peaks of the
$x$ and $y$ scans (including during any pause between the two scans).
A horizontal orbit drift between the $x$-scan peak (where the beams are
perfectly centred horizontally by definition) and the $y$-scan peak (where
they may no longer be), would lead to an underestimate of the visible
interaction rate at the peak of the $y$ scan, and hence of y in
Eq. (<ref>).
Similarly, a vertical orbit drift causes an underestimate
of the peak interaction rate in the $x$ scan.[Strictly speaking,
what matters is the cumulative effect on the collision rate of the change
in horizontal (vertical) beam separation, relative to that at the peak
of the $x$ ($y$) scan, from the beginning to the end of the $y$ ($x$) scan.
If the orbit drift rate is constant in time during the individual $x$ and $y$
scans, the cumulative effect on  is equivalent to that evaluated
from the  scan peaks alone.]
Approximate corrections for out-of-plane drifts were made using the
differences in horizontal and vertical orbit drifts
$\delta x^0$ and $\delta y^0$ between the peaks of the $x$ and $y$ scan curves:

\begin{eqnarray*}
\delta x^0 & = &
\delta^x_1(t^0_y)-\delta^x_2(t^0_y)
-\delta^x_1(t^0_x)+\delta^x_2(t^0_x) \ ,\\
\delta y^0 & = &
\delta^y_1(t^0_x)-\delta^y_2(t^0_x)
-\delta^y_1(t^0_y)+\delta^y_2(t^0_y) \ ,
\end{eqnarray*}

where the central (zero nominal separation) points of the $x$ and $y$ scan
curves occur at times $t^0_x$
and $t^0_y$. It can be shown <cit.> that the
visible cross-section defined in Eq. (<ref>) and corrected for
out-of-plane orbit drifts is  given by

\begin{equation}\nonumber
\sigmavis = 4\pi\capsigx\capsigy\frac{{\cal G}_x(\Delta x^\mathrm{max}){\cal G}_y(\Delta y^\mathrm{max})}{{\cal G}_x(\delta x^0)+{\cal G}_y(\delta y^0)} \ ,
\end{equation}

where ${\cal G}_x(\Delta x)$  and ${\cal G}_y(\Delta y)$ denote the functions
of nominal separation fitted to the normalised vdM scan curves
$\mubvis(\Delta x)$ and $\mubvis(\Delta y)$, and
$\Delta x^\mathrm{max}$ and $\Delta y^\mathrm{max}$ are the fitted nominal
beam separations at the peaks of these functions, i.e.${\cal G}_x(\Delta x^\mathrm{max})=\mubvismaxi{x}$ and
${\cal G}_y(\Delta y^\mathrm{max})=\mubvismaxi{y}$. This formalism corrects
for both out-of-plane orbit drifts and any imperfect initial alignment
in the non-scanning plane.

A third effect is associated with the potential change in  due to
drifts in the $y$ (i.e. non-scanning) plane during the $x$ scan, and similarly
for  due to drifts in $x$ during the $y$ scan. This effect is distinct
from the effects of out-of-plane drifts on  discussed above. However,
since the beams were reasonably well-centred on each other before each scan
started (as ensured by the scan protocol), these effects are negligible for
the orbit drifts observed in the non-scanning plane during on-axis scans.

Large orbit drift corrections were seen for scan S4 in 2015
(shifting  by 2.6%), S1 in 2017 (1.1%) and S4 in 2018 ($-0.8$%),
mainly due to drift in the horizontal plane. All other scans
have orbit drift corrections at the level of $\pm 0.5$% or less. Applying the
corrections always improved the scan-to-scan consistency within each year. The
baseline corrections were evaluated using the DOROS BPM system, and using
the arc BPMs gave similar results. The `Orbit drift correction'
uncertainties in Table <ref> were calculated as the fractional
scan-averaged difference between the  values determined using
the DOROS and arc BPM corrections, and are smaller than 0.1% in all years.

Short-term movements of the beams during the 30 s duration of a single vdM
scan step may lead to fluctuations in the luminosity, distorting the measured
scan curve. The potential effect of this beam position jitter was characterised
by studying the distribution of the individual arc BPM beam position
extrapolations, which were sampled every 5–7 s,  i.e. several times
per scan step. All measurements in a single $x$ or $y$ scan were fitted as a
function of time to a second-order polynomial, and the RMS of the
residuals of the individual measurements with respect to this fit were
taken to be representative of the random beam jitter. Data from the $y$ scan
(where the beams are stationary in $x$) were used to measure the jitter in the
$x$-direction, and vice versa,
giving RMS values in the range 0.5–1.0 $\mu$m, typically larger in the
horizontal plane than the vertical.
One thousand simulated replicas of each real data scan were created,
applying Gaussian smearing to the beam positions (and hence the separation) at
each step according to the RMS of the arc BPM fit residuals, coherently
for the data from all colliding bunch pairs. The RMS of the  values
obtained from these fits, averaged over all scans in a year, is typically
around 0.2%. No correction was made for this effect, but the RMS
was used to define the `Beam position jitter' uncertainty in Table <ref>.

§.§ Non-factorisation effects

The vdM formalism described by Eqs. (<ref>–<ref>) assumes that
the particle density distributions in each bunch can be factorised into
independent horizontal and vertical components, such that the term
$1/2\pi\capsigx\capsigy$ in Eq. (<ref>) fully describes the overlap
integral of the two beams. Evidence of non-factorisation was clearly
seen during Run 1 <cit.>, especially when dedicated beam
tailoring in the LHC injector chain was not used. Such beam tailoring, designed
to produce Gaussian-like beam profiles in $x$ and $y$,
was used for all  vdM scan sessions during Run 2, and information
derived from the distribution of primary collision vertices reconstructed
by the ATLAS inner detector in both on- and off-axis scans was used
to constrain the possible residual effect of non-factorisation on the determination, following the procedure of Ref. <cit.>.

In more detail, combined fits were performed to the beam-separation dependence
of both the LUCID luminosity measurements and the position, orientation and
shape of the luminous region, characterised by the three-dimensional (3D)
spatial distribution of the primary collision vertices formed from tracks
reconstructed in the inner detector (i.e. the beamspot) <cit.>.
Since this procedure requires a large number of
reconstructed vertices per scan step, it was carried out for only a
handful of colliding bunch pairs, whose tracking information was read out
at an enhanced rate. Non-factorisable single-beam profiles (each composed of a
weighted sum of three 3D Gaussian distributions $G(x,y,z)$ with arbitrary
widths and orientations)
were fitted to these data. These profiles were then  used in simulated vdM
scans to determine the ratio $R$ of the apparent luminosity scale which
would be extracted using the standard factorisable vdM formalism
to the true luminosity scale determined from the full 3D beam overlap
integral. This procedure was carried out both for each on-axis $x$–$y$
scan in each vdM dataset, and with fits combining each off-axis (and in 2016,
diagonal) scan with the closest-in-time on-axis scan. The resulting
$R$ values for the 2016 and 2017 datasets are shown in Figure <ref>.
In general, the results from on-axis and combined fits are similar,
although there is significant scatter between bunches and scans.

Non-factorisation correction factors $R$ for several
colliding bunch pairs (BCID) and vdM scan sets in the (a) 2016 and (b) 2017
vdM sessions. The results were extracted from combined fits to the vdM
luminosity scan curves and reconstructed primary vertex data, using either
on-axis scans alone or combined fits to an on-axis scan and an off-axis or
diagonal scan. The uncertainties are statistical.
The dashed horizontal lines show the error-weighted
mean corrections,
and the yellow bands the uncertainties assigned from the RMS of the measured
$R$ values. The results from the different scans for each colliding bunch pair
are slightly offset on the horizontal axis for clarity.

The ratio $R$ was used to correct the visible cross-section measured by the
standard factorised vdM analysis for the measured non-factorisation:
$\sigviscorr=\sigmavis/R$. Since the evaluation of $R$ only sampled a small
fraction of the colliding bunch pairs,
the error-weighted average of $R$ over this subset of bunches was
taken as representative of the complete set, and applied to
the bunch-averaged , giving corrections of $R=1.006\pm 0.003$ for
2016, $0.998\pm 0.001$ for 2017 and $1.003\pm 0.003$ for 2018, where the
uncertainties were defined as the RMS of the individual $R$ values for all
available bunches and scans in each year. In 2015, the same method gave
a central value of $R=0.995$ but with a poor fit to the offset scan
S3-off. An alternative method, fitting a non-factorisable function of both
$x$ and $y$ to the $x$ and $y$ scan curves simultaneously, but without the
use of beamspot data, gave a central value of $R=1.006$. For the 2015 dataset,
a value of $R=1.000\pm 0.006$ was used, i.e. no correction but an uncertainty
that spans the range of central values from the two methods.

§.§ Beam–beam interactions

The mutual electromagnetic interaction between colliding bunches produces
two effects that may bias their overlap integral, and that depend on the beam
separation: a transverse deflection that induces a non-linear distortion of the
intended separation, and a defocusing of one beam by the other that not only
modulates the optical demagnification from the LHC arcs to the interaction
point, but also modifies the shape of the transverse bunch profiles.

The beam–beam deflection can be modelled as a kick produced by a small dipole
magnet, i.e. for a horizontal scan as an angular deflection $\theta_{x,i}$ that
results in a
horizontal beam displacement or orbit shift i of each beam $i$ at the
IP. In the round-beam approximation, the deflection of a beam-1 bunch during a
horizontal on-axis scan is given by <cit.>

\begin{equation}\label{e:bbang}
\theta_{x,1}=\frac{2r_{p} n_2}{\gamma_1\Delta x}\left[1-\exp(-\Delta x^2/(2\capsigx^2))\right];\ \theta_{x,2} \approx -\theta_{x,1}\ ,
\end{equation}

where $r_{p}=e^2/(4\pi\epsilon_{0}\,m_{p}\,c^2)$ is the classical proton radius,
$\gamma_1=E_1/m_{p}$ is the Lorentz factor of the beam-1 particles
with energy $E_1$ and mass $m_{p}$,
$n_2$ is the charge of the beam-2 bunch, $\Delta x$ is the nominal beam
separation produced by the steering corrector magnets, and  is the
transverse convolved beam size defined in Eq. (<ref>). The induced
single-beam displacement at the interaction point is proportional to
$\theta_{x,i}$, and is given by

\begin{equation}\nonumber
\dxbb{i}=\theta_{x,i}\frac{\beta^*_x}{2\tan(\pi Q_x)};\
\dxbb{2}\approx -\dxbb{1} \ \ ,
\end{equation}

where $\beta^*_x$ is the value of the horizontal $\beta$-function at the IP and
$Q_x$ is the horizontal tune.[The tune of a storage ring is defined
as the betatron phase advance per turn, equivalent to the number of betatron
oscillations over one full ring circumference.]  Similar formulae apply to a
vertical scan. The beam–beam deflection at each scan point, and its impact on
the corresponding beam separation, can be calculated analytically from the
measured bunch currents and  values using the Bassetti–Erskine
formula for elliptical beams <cit.>. An example for one
colliding bunch pair in the 2017 scan S1X (i.e. the $x$-scan of S1)
is shown in Figure <ref>(a).
The change of separation is largest for $|\Delta x|\approx 200\,\mu$m, where it
amounts to almost $\pm 2\,\mu$m, and decreases to around $\pm 1\,\mu$m in the
tails of the scan.
Beam–beam deflections were taken into account on a bunch-by-bunch basis, by
correcting the nominal separation at each scan step by the calculated orbit
shifts for both beams

\begin{equation}\nonumber
\Delta x^\mathrm{corr}=\Delta x+\dxbb{1}-\dxbb{2}\ ,
\end{equation}

then refitting the vdM-scan curves and updating .
This orbit-shift correction
causes a 1.7–2.2% increase in , depending on the bunch parameters.

(a) Calculated change in beam separation due to the
beam–beam deflection (orbit shift) and (b) optical-distortion correction, as
functions of the nominal beam separation in the horizontal plane, for bunch
slot (BCID) 1112 in the horizontal scan 1 of the 2017 vdM session. The nominal
values are shown by the black points and curve. The solid and dashed lines
illustrate the impact of varying the $\beta^*$ values by $\pm 10$% or the
non-colliding tunes $Q$ by $\pm 0.002$, in each case for both beams and
both planes simultaneously.

In the Run-1 luminosity calibrations <cit.>, the
defocusing effect was estimated using the MAD-X optics code <cit.>,
giving a negative optical-distortion correction of 0.2–0.3% in , and
thereby cancelling out a small fraction of the orbit-shift correction.
However, this calculation implicitly neglected the fact that the gradient of
the beam–beam force is different for the particles in the core of the bunch
than it is for those in the tails.
Recent studies <cit.> using two independent
multiparticle simulation codes, B*B <cit.> and COMBI <cit.>,
have shown that the linear-field approximation used by MAD-X is inadequate,
and that the actual optical-distortion correction is larger than previously

Starting from the assumption of two initially round Gaussian proton bunches,
with equal populations ($n_1=n_2$) and sizes
($\sigma_{x,1}=\sigma_{x,2}=\sigma_{y,1}=\sigma_{y,2}$), that collide at a single
interaction point with zero crossing angle and the same $\beta$-function
($\beta^*_x = \beta^*_y = \beta^*$), Ref. <cit.> demonstrates that the
optical-distortion correction at each nominal beam separation $\Delta x$ can be
characterised by a reduction in luminosity $\lref/L$ parameterised as a function
of $\Delta x$, the non-colliding tunes[The non-colliding tunes refer
to the tunes in the absence of collisions at any of the four LHC IPs.]
$Q_{x,y}$ and the round-beam equivalent beam–beam parameter :

\begin{equation}\nonumber
\frac{\lref}{L}\left(\Delta x\right)  =  f(\Delta x, Q_x, Q_y, \xir)\ ,
\end{equation}

\begin{equation}\label{e:xir}
\xir  =  \frac{r_p\bar{n}\beta^*}{2\pi\gamma\capsigx\capsigy} \ .
\end{equation}

At each given nominal separation $\Delta x$, $\lref$ is the luminosity in the
absence of (or equivalently, after correction for) optical-distortion effects,
$L$ is the uncorrected (i.e. measured) luminosity,
$\bar{n} = (n_1 + n_2)/2$ is the average number of protons per bunch,
$r_p$ is the common classical particle radius, and $\gamma$ is the common
Lorentz factor. For the Run 2  vdM scans, the parameter  varies from
$3\times 10^{-3}$ to $4\times 10^{-3}$. Reference <cit.> provides a
polynomial parameterisation of the correction $\lref/L$ obtained from the full
B*B and COMBI simulations, valid for the range of beam parameters typical
of LHC vdM scans.

An additional complication arises from the effects of collisions at
`witness IPs', i.e. at interaction points other than that of ATLAS.
Studies with COMBI simulations <cit.>
showed that the effect on the beam–beam corrections
can be adequately modelled by an ad-hoc shift $\Delta Q_{x,y}$ in the
non-colliding tune values that are input to the single-IP parameterisation

\begin{equation*}
\Delta Q_{x,y}= p_1(\nwit)\,\xir\ ,
\end{equation*}

that accounts for the average beam–beam tune shift induced by collisions at
interaction points other than the one where the scan is taking place.
Here,  is the number of witness IP collisions for the colliding bunch
pair in question, incremented by one half for each additional collision that
each separate bunch experiences around the LHC ring, and $p_1(\nwit)$ is
a first-order polynomial function of .
Since all bunches that collide in ATLAS also collide in CMS, $\nwit\geq1$.
Depending on the LHC filling pattern, some bunches also experience collisions
in either LHCb or ALICE, leading to some bunch pairs with
$\nwit=1.5$ or $\nwit=2$, especially in 2018.

The resulting optical-distortion correction for one particular colliding bunch
pair in the 2017 scan S1X is shown in Figure <ref>(b); the luminosity
is reduced by 0.5% at the peak and almost 2% in the tails, depending slightly
on the $\beta^*$ and tune values. This correction was applied to the measured
luminosities (and hence  values) at each scan point, reducing
by around 1.5%.
The values of the orbit-shift, optical-distortion and total beam–beam
corrections, averaged over all colliding bunch
pairs, are shown for each scan in each year in Figure <ref>.
The revised optical-distortion correction cancels out a much larger fraction of
the orbit-shift correction than predicted by the original linear treatment.

Bunch-averaged corrections to  due to
the orbit shift and optical distortion caused by the mutual electromagnetic
interaction of the two beams, calculated for all Run 2  on-axis vdM scans,
showing the two effects separately, and their total. The error bars show
the uncertainties of the total correction.

The systematic uncertainty affecting the magnitude of the beam–beam
corrections <cit.> is dominated by the tunes.
An uncertainty of 0.002 is assigned to the non-colliding tunes, correlated
between beams and planes; an additional  uncertainty of 0.001 in the mean tunes
input to the parameterisation accounts for the approximate modelling of the
effect of witness-IP collisions on the unperturbed-tune[The unperturbed
tunes are the values of the horizontal and vertical tunes with the beam–beam
interaction `switched off' at the IP where the scans are taking place
(whether or not the beams are colliding at other IPs). Physically, these
correspond to the mean tune values that would be measured, for the bunch pair
under study, before the beams are brought into collision at this IP.] spectra.
The uncertainty in the actual value of $\beta^*$ (which is difficult to measure
precisely) is conservatively taken as  10%, uncorrelated between the two beams
but correlated between $x$ and $y$; it directly translates into an uncertainty
in the value of  defined in Eq. (<ref>). A number of additional
uncertainties, detailed below,
arise due to the assumptions made in the evaluation of the
combination of the orbit-shift and optical-distortion corrections <cit.>.
An uncertainty of 0.1% results from
potential non-Gaussian tails in the initial
unperturbed transverse beam profiles. Other effects, such as the departure from
round beams (i.e. $\capsigx\neq\capsigy$), unequal transverse beam sizes, limitations in the parameterisation of optical distortions, uncertainties in the
modelling of witness-IP effects,
magnetic lattice non-linearity, and the effects of a residual beam
crossing angle, each lie below 0.1%.
Most of the uncertainties are either correlated
or anticorrelated between the orbit-shift and optical-distortion corrections;
their effects were therefore evaluated taking this into account.
The total beam–beam uncertainty in  in
Table <ref> is less than 0.3% for all years.

§.§ Emittance growth corrections

The determination of  from Eq. (<ref>) implicitly assumes
that the convolved beam sizes  (and therefore the transverse
emittances <cit.> of the two beams)
remain constant both during a single $x$ or
$y$ scan and between the peaks of the two scans. The result
may be biased if the beam sizes (and hence also the peak interaction
rates) change during or between $x$ and $y$ scans.

The evolution of the emittances during the vdM scan sessions was studied by
comparing the values of  obtained from the different on-axis scans
in each vdM fill. These studies showed that  generally
increases through the fill at rates of 0.2–0.6 $\mu$m/hour, whereas
decreases at around 0.8–1.3 $\mu$m/hour, due to
synchrotron-radiation damping <cit.>.
The peak interaction rates (after correcting for out-of-plane
orbit drifts as discussed in Section <ref>) also increase with
time, consistent with the larger change of emittance in the $y$ plane
(as a decrease in emittance should correspond to an increase in peak rate).
The effect of emittance growth on the estimate of  from
a single $x$ or $y$ scan was studied by fitting simulated scan curves,
described by Gaussian distributions whose widths changed during the scan
according to the observed rates. The largest bias was found to be 0.03%,
with typical values being an order of magnitude smaller, and this effect was
thus neglected.

Much larger potential biases in  arise from
the evaluation of the $x$- and $y$-related
quantities in Eq. (<ref>) at different times. Making
the time-dependence explicit, Eq. (<ref>) can be rewritten as

\begin{equation*}
\sigmavis(t_x,t_y)=\frac{1}{2}\left( \mubvismaxi{x}(t_x)+\mubvismaxi{y}(t_y) \right)\cdot 2\pi\capsigx(t_x)\capsigy(t_y)\ ,
\end{equation*}

where the $x$-related quantities are evaluated at time $t_x$ and the $y$-related
quantities at time $t_y$. These quantities can be translated to a common
time $\tmid=(t^0_x+t^0_y)/2$ midway between the peaks of the $x$ and $y$ scans
at $t^0_x$ and $t^0_y$, using
linear fits of the evolution of  and peak rates with time
(this procedure is only possible for
fills which have more than one on-axis scan). The emittance growth correction
can then be defined as

\begin{equation}\nonumber
\cee = \frac{\sigmavis(\tmid,\tmid)}{\sigmavis(t^0_x,t^0_y)}-1 \ ,\\ \nonumber
\end{equation}

correcting the measured visible cross-section for the bias caused by evaluating
the $x$- and $y$-scan quantities at different times:
$\sigviscorr=(1+\cee)\,\sigmavis$. Since the visible
cross-section should not depend on the choice of ,  was also
evaluated by evolving all quantities to $t=t^0_x$ or $t=t^0_y$, taking the
largest difference as the uncertainty in  due to the linear evolution
model. This uncertainty also accounts for the small inconsistencies observed
between the evolution of the measured  and that predicted from the
measured evolution of .

Emittance growth correction  for each vdM scan
in fills with more than one scan, shown as a function of the elapsed time
between the peaks of the $x$ and $y$ scans. The corrections
were evaluated by determining the rate of change
of emittances during the sequences of scans given in the legend, dividing
the 2017 scans into two groups. The error bars show the total uncertainties,
including the statistical uncertainties from the linear fits and the systematic
uncertainty due to the mismatch between the  and peak-rate evolution.

The corrections for each scan recorded in a fill with more than one
on-axis scan are shown as a function of the elapsed time between
the $x$ and $y$ scan peaks in Figure <ref>, together with the
systematic uncertainty from the fit model combined with the statistical
uncertainties from the linear fits. The corrections increase ,
typically by 0.15–0.3%, except for scan S1 in 2017, where there was a delay
between the $x$ and $y$ scans, and hence a larger correction of 0.39%. Although
all four 2017 scans were performed in the same fill, there was a six-hour
gap between S1+S2 and S4+S5. The two pairs of scans have different
emittance growth rates, so they were analysed separately. Scans S4 in 2015 and
S6 in 2016 were performed in separate LHC fills (see Table <ref>)
with only a single on-axis scan, so they were assigned the average correction
for other scans in that year. In all cases, the corrections were calculated
using the bunch-averaged  and , but a bunch-by-bunch
analysis showed very similar results, the emittance growing at similar rates
in all bunches. The final uncertainties due to emittance growth corrections
shown in Table <ref> are smaller than 0.1% for all years.

§.§ Length scale determination

The  measurements require knowledge of the length scale, i.e.the actual beam displacements (and hence beam separation) produced by
the settings of the LHC steering magnets intended to produce a given nominal
displacement. This was determined using length scale calibration (LSC) scans,
separately for each beam and plane ($x$ and $y$). In each scan, the target
beam being calibrated was moved successively to five equally spaced positions
within $\pm 3\signom$, and its position measured using the
luminous centroid (or beamspot) position fitted from primary collision vertices
in the ATLAS inner detector when the two beams were in head-on collision. Since
the alignment of the inner detector is very well understood
<cit.>, the beamspot
gives a measurement of the beam displacement that is accurate at the 0.1%
level over the scanning range. In practice, the requirement for the two
beams to be in head-on collision was satisfied by performing a three-point
miniscan of the non-target beam around the target beam nominal position at
each step, fitting the resulting curve of luminosity vs. beamspot position,
and interpolating the beamspot position to that of maximum luminosity and beam
As shown in Figure <ref>(a), the slope of a linear fit of the beamspot
position vs. the nominal position gives the linear length scale $M_1$, i.e. the
ratio of actual to nominal beam movement for each beam and plane (the offset of
this fit depends on the relation between the LHC beam and ATLAS coordinate
systems). Orbit drifts can influence the measured
values of $M_1$, and were corrected using arc BPM data, interpolating
the difference in beam positions measured directly before and after each
scan linearly in time through the scan, in a procedure similar to that
described for the vdM scans in Section <ref>.

(a) Length scale calibration for the $y$ direction
of beam 2 in the 2017 measurement, showing the beamspot position vs. the nominal
displacement of the target beam,
together with a linear fit giving the slope parameter $M_1$. The
lower plot shows the fit residuals, with the error bars indicating the
statistical uncertainties of the beamspot positions. These residuals
are discussed further in Section <ref>. (b) Measurements of the
linear length scale $M_1$ for each beam and plane measured in the
calibration for each year of data-taking.

The four length scales for B1X (i.e. beam 1 in the $x$-direction),
B1Y, B2X and B2Y were measured in turn in the LSC
scans for each year, which were always performed with the same machine
optics as, and close in time to, the vdM scans. The resulting values of $M_1$
for all four quantities are shown in Figure <ref>(b), and are always
within $\pm 0.4$% of unity. However, apart from B1Y, the length scales
differ significantly between years, despite the same nominal machine
configuration being used. In 2015–17, all four scans were performed from
negative to positive displacement, meaning that beam 1 moved in the same
direction as in the vdM scans, but beam 2 moved in the opposite direction.
In 2018, the beam-2 LSC scans were performed from positive to negative
displacement, so both beams were calibrated by moving them in the same
direction as in the vdM scans.

The measured  values from
Eq. (<ref>) must be corrected by the average of the beam-1 and
beam-2 length scales $M_1$ in each plane, leading to a correction  to the
visible cross-section $\sigviscorr=\lsp\sigmavis$, where the length scale
product  is given by

\begin{equation*}
\lsp=(M_1^{x,1}+M_1^{x,2})(M_1^{y,1}+M_1^{y,2})/4 \ ,
\end{equation*}

and $M_1^{x|y,i}$ is the measured linear length scale for the $x$ or $y$ plane
and beam $i$. The statistical uncertainty from the beamspot position
measurements is shown as the `Length scale calibration' uncertainty in
Table <ref>, and is below 0.1%. A further uncertainty of 0.12%
arises from uncertainties in the absolute length scale of the inner detector,
determined from simulation studies of various realistic misalignment scenarios
as described in Ref. <cit.>. The studies were updated to use
scenarios appropriate to the Run 2 detector, including the impact of the
precise measurements from the innermost pixel layer, the IBL.

§.§ Magnetic non-linearity

The fit residuals shown in the lower panel of Figure <ref>(a) suggest
that, in this particular scan, the actual beam movement is not perfectly
linear with respect to the
nominal position, at the level of $1\,\mu$m in $\pm 300\,\mu$m of displacement.
The residuals from some of the other  LSC scans show similar hints of
systematic non-linear behaviour; some are more scattered, and some appear
consistent with a purely linear dependence. The  LSC data from 2012
are also suggestive of non-linear behaviour <cit.>.
More recent data are available to study these effects further.
For five days in October 2018,
the LHC delivered $pp$ collisions at  in support of the forward-physics
programme. As part of this run, a vdM scan session, including a length
scale calibration, took place with $\beta^*=11$ m, a configuration
in which the transverse beam sizes (and hence scan ranges) are around three
times larger than in the vdM optics used at .
As well as the length scale calibration,
a total of seven $x$–$y$ vdM scans were performed over four fills spanning a
three-week period, as shown in the top part of Table <ref>. The
residuals from the length scale calibration are shown in
Figure <ref>, and show a clear deviation from linearity,
of up to $\pm 3\,\mu$m over a $\pm 900\,\mu$m scan range. The shape of
the residual curve is approximately inverted for beam 2 compared to beam 1
in both planes. As the two beams were moved in opposite directions during the
LSC, this suggests the non-linearity may come from hysteresis effects in
the steering corrector magnets.

Summary of the  vdM scan fills used in the study
of magnetic non-linearity, showing the dates, LHC fill numbers, total numbers of
bunches in each beam () and colliding in ATLAS (), and the individual
scans performed in each fill. In the first group of scans performed in 2018,
each scan consisted of an $x$–$y$ vdM scan pair (S1, S3 etc.), an off-axis scan
(S2-off) or a length scale calibration scan (LSC). The second group
of scans was performed during the LHC beam test in October 2021, and individual
$x$ or $y$ scans are listed separately, as parallel (par), separation (sep)
or single-beam (B1 or B2) scans.
Date     Fill             Scans

14/10/2018     7299     152     150     S1, S2-off, S3

14/10/2018     7300     152     150     LSC, S4

5/11/2018     7406     152     150     S5

5/11/2018     7407     152     150     S6, S7-off

28/10/2021     7516     4     2     S1xsep, S2xpar, S3ypar, S4ysep, S5ypar, S6xB1, S7xB2, S8ypar

30/10/2021     7524     3     0     S1xpar, S2xpar, S3xsep, S4xpar, S5ypar, S6ypar, S7ysep, S8ypar

30/10/2021     7525     3     3     S1xpar, S2xsep, S3ypar, S4ysep, S5yB1, S6yb2, S6xB1, S7xB2

Residuals of the beamspot position with respect
to a linear fit of the beamspot position vs. nominal displacement of the target
beam in the length scale
calibration of the 2018  vdM session, for (a) the $x$ and (b) the $y$
plane, for beam 1 (blue open points) and beam 2 (red filled points).
The error bars show the statistical uncertainties from the beamspot
position measurements.

The beamspot positions measured during the  LSC scans provide
unambiguous evidence of non-linearity, but have limited granularity
(only five points per scan) and do not address reproducibility, as only
one scan was performed per beam/plane. The multiple vdM scans per session
have 25 points per scan and so can potentially address these issues, but since
the beams move in opposite directions, the beamspot position remains
approximately stationary, only moving slightly if the two beams are of
unequal sizes. The DOROS BPMs measure the displacements of each beam separately
during vdM scans, and can be used to study magnetic non-linearity effects.
However, they suffer from short-term variations in their calibration,
and must also be corrected for the effects of beam–beam deflections.

The differences between the beam position at the interaction point determined
from the uncalibrated DOROS BPM measurements, and the nominal
beam-1 displacements, are shown  during the  $x$ scans from 2018 in
Figure <ref>(a). The residuals are shown after subtracting an
offset from the BPM
such that the residual with the beam at the nominal head-on position,
immediately before the start of each scan, is zero. A significant slope is
visible, indicating a miscalibration of the length scale of the BPM.
The residuals are slightly negative at the nominal central point of the scan,
suggesting magnetic hysteresis. The curves also have an `S' shape,
due to the beam–beam deflection already discussed in
Section <ref>. As well as the true beam displacement at the
interaction point, the DOROS BPMs also measure an additional apparent
displacement from half the angular kick
$\theta_{x,i}$ projected over the distance $L_\mathrm{doros}=21.7$ m between
the IP and the BPM.[The factor of one-half arises because in the new
closed orbit, both the incoming and outgoing beams are deviated from their initial trajectories by $\theta_{x,1}/2$, giving a total angular deflection of
$\theta_{x,1}$ at the IP.]  With $\theta_{x,i}$ given by
Eq. (<ref>), the apparent displacement i of beam $i$ at the IP
as measured by the DOROS BPMs is

\begin{equation}\label{e:bbdeld}
\dxbbd{1}=\theta_{x,1}\left(\frac{\beta^*}{2\tan(\pi Q_x)}
+\frac{L_\mathrm{doros}}{2}\right);\ \dxbbd{2}=-\dxbbd{1} \ .
\end{equation}

The expected apparent horizontal beam–beam displacements for beam 1 in
these scans, calculated
using the measured  and beam currents, are shown in
Figure <ref>(b) and reach $\pm 15\,\mu$m at $\pm 250\,\mu$m
single-beam displacement (corresponding to a separation $\Delta x=500\,\mu$m).
The displacements for the off-axis scans S2 and S7 are smaller, and were
calculated using a generalisation of Eq. (<ref>) that also takes into
account the constant beam separation in the non-scanning plane.

Details of the analysis of beam-1 DOROS BPM
residuals in the $x$ scans
of the 2018  vdM scan session: (a) residuals between the uncalibrated
BPM measurements and the nominal beam displacement as a function of the
nominal displacement; (b) apparent beam–beam-induced displacement at the IP as
seen by the DOROS BPMs, calculated using the measured beam parameters.

The miscalibration and offset of the DOROS BPM measurement visible in
Figure <ref>(a) can be corrected by replacing the initial
DOROS estimate  of the beam position at the IP xi by
$\dcal{x}{i}\doros{x}{i}-\dofs{x}{i}$, where $\dcal{x}{i}\approx 1$
is the DOROS length scale and $\dofs{x}{i}$ the offset at the central scan
point.  Using the calibrated
DOROS measurement, together with the nominal beam displacement xiscaled by its length scale $M_{1,i}$ and the estimate of the apparent
beam–beam deflection i from Eq. (<ref>) scaled by a parameter
, the residual between the calibrated DOROS measurement and the
nominal beam displacement for beam $i$ at a particular scan point is given

\begin{equation}\label{e:dresid}
\resid{doros}{x}{i}=\dcal{x}{i}\doros{x}{i}-\dofs{x}{i} -M_{1,i}\setpos{x}{i}-\bbds\,\dxbbd{i} \ .
\end{equation}

These residuals were used to define a $\chi^2$ summed over all the
scan points in a single scan or set of scans for one beam and plane:

\begin{equation}\label{e:chidoros}
\chi^2_\mathrm{doros}=\sum \left(\frac{\resid{doros}{x}{i}}{\sdd\,\sigma_\mathrm{doros}}\right)^2\ ,
\end{equation}

where the statistical uncertainties in the DOROS BPM measurements
$\sigma_\mathrm{doros}$ were determined from the spread of individual
measurements over the duration of one LB, and  is a scale factor with
nominal value $\sdd=1$. The
$\chi^2$ was minimised, allowing the values of xi, xi and
to be determined. The length scale $M_1$ cannot be determined
from this fit (as it is almost degenerate with xi), but was determined
within the same framework by using the residuals between the beamspot and
nominal beam positions in the LSC scan to define an additional
$\chi^2$ term constraining $M_1$:

\begin{equation}\label{e:chilsc}
\chi^2_\mathrm{lsc}=\sum \left(\frac{\xbs{}-M_{1,i}\setpos{x}{i}}{\exbs{}}\right)^2\ ,
\end{equation}

where  is the beamspot position measurement at maximum beam overlap
derived from each LSC miniscan and  its uncertainty,
and the sum is taken over the five LSC scan points.[This $\chi^2$
minimisation to determine $M_1$ is equivalent to the linear fit shown in
Figure <ref>(a).]
A combined fit to all seven vdM scans and the LSC was performed separately for
each beam and plane to determine xi, xi, $M_1$ and .
Separate values of xi and xi were fitted in each scan,
and these were found to differ scan-to-scan with RMS values of 0.3–0.4%
(the DOROS BPM responses are not expected to be stable in time, due to
temperature variations in their electronics). For each beam and plane,
a common  value was fitted for all scans, and these values differed
from unity by up to 17%.

The resulting DOROS residuals from Eq. (<ref>)
are shown for each beam and plane in Figure <ref>, together with the
beamspot residuals from the LSC scan that were also shown in
Figure <ref>.  In both $x$ and $y$ planes for beam 2,
the residuals from all scans are similar, and close to those from
the LSC scan, suggesting a non-linearity that is largely reproducible between
scans, and present whether the beam displacements are measured with DOROS BPM
or beamspot data. The situation for beam 1 appears more complex: the DOROS
broadly follow the beamspot data, but several scans exhibit residual S-shaped
wiggles even after subtracting the scaled beam–beam displacement, suggesting
that the magnitude of the apparent beam–beam displacement has been
overestimated or underestimated.
Allowing the value of  to be different for each
scan (as well as for each plane) significantly improved the fits, giving
residuals for beam 1 that resemble the beam-2 curves, although inverted
due to the opposite scan direction. However, the fitted  values
scatter between 0.5 and 1.2, typically being smaller for beam 1 than for beam 2.
This large range of fitted $\alpha$ values
suggests that although the separation-dependence of the beam–beam
displacement seen by
the DOROS BPMs is well predicted by Eqs. (<ref>) and (<ref>),
its magnitude is not.

Residuals between the beam displacement
at the interaction point inferred from the calibrated DOROS BPM measurement
corrected for the beam–beam-induced displacement,
and the nominal beam displacement, during vdM scans at  in 2018,
using a fit with a linear magnetic model and a single beam–beam deflection
scaling parameter  per beam and plane. The four
plots show (a) B1X, (b) B1Y, (c) B2X and (d) B2Y. The residuals
for the seven scans are shown separately by the different coloured markers.
The black stars show the differences between the beamspot position and
the nominal beam displacement measured in the length scale calibration scan and
plotted on the same axes.

The non-linearity between the nominal and actual beam displacements i and i was parameterised as

\begin{equation}\label{e:lscnonlin}
\xpos{i}=M_{1,i}\xset{i}+M_{2,i}r_i^2+M_{3,i}r_i^3\ ,
\end{equation}

where $M_{1,i}$ is the linear length scale, and $M_{2,i}$ and $M_{3,i}$ are the
coefficients of quadratic and cubic terms in the reduced beam separation
$r=\xset{i}/x_\mathrm{max}$, where $x_\mathrm{max}=850\,\mu$m is the maximum
beam displacement in a  scan. The corresponding non-linear terms
were added to Eqs. (<ref>) and (<ref>), and the fits repeated,
also allowing  to be different in each scan. The resulting residuals
are typically smaller than $0.5\,\mu$m and without significant structure,
demonstrating that the non-linearity seen in both DOROS BPM and beamspot
measurements can be reasonably well parameterised by Eq. (<ref>).
The fitted values of $M_1$, $M_2$ and $M_3$ for both beams and planes are
shown in Table <ref>, together with the $M_1$ values from linear
fits to the LSC scan only, analogous to that shown in Figure <ref>.
The significant quadratic terms of $|M_2|\approx 4\,\mu$m, with opposite signs
for the two beams, account for the symmetric distortions seen in
Figure <ref>. The cubic $M_3$ terms account for any distortions that
are asymmetric between positive and negative beam displacements, and their
fitted values are all smaller than $1\,\mu$m and consistent with zero.
However, the $M_3$ and
$M_1$ terms are anticorrelated, and the $M_1$ values also change
slightly between the linear and non-linear fits.

Fitted values of the length scale $M_1$ from
a linear fit to the LSC scan data only (left column) and the parameters
$M_1$, $M_2$ and $M_3$ from the non-linear
fit to vdM and LSC scan data (right columns), for each beam and plane
in the 2018  vdM scan session.
$M_1$ (linear LSC)     $M_1$     $M_2$ [$\mu$m]     $M_3$ [$\mu$m]

B1X     $1.0044\pm 0.0003$     $1.0046\pm 0.0006$     $3.7\pm 0.2$     $-0.2\pm 0.5$

B1Y     $1.0064\pm 0.0003$     $1.0067\pm 0.0003$     $4.5\pm 0.1$     $-0.3\pm 0.2$

B2X     $1.0022\pm 0.0003$     $1.0024\pm 0.0006$     $-4.2\pm 0.2$     $-0.1\pm 0.5$

B2Y     $1.0012\pm 0.0003$     $1.0014\pm 0.0003$     $-4.2\pm 0.2$     $-0.2\pm 0.2$

The effect of the non-linear terms on the length scale product was estimated
by modelling the scan curves as single Gaussian functions with RMS equal to the
averaged over all scans, and calculating the change in the
integrals of Eq. (<ref>) from the distortions in $\Delta x$ and
$\Delta y$ when using the non-linear instead of the linear length scale. For the
2018  scans  changes from $1.0071\pm 0.0003$ to $1.0075\pm 0.0004$,
a change of only 0.04%. Although the $M_2$ terms are large, they have no
effect on , because e.g. a decrease of the integral for $\Delta x<0$ is
compensated by an equal increase for $\Delta x>0$, and only the small changes
in $M_1$ and $M_3$ have any effect.

Since the non-linear effects seen in the  scans were unexpected, two
additional studies were made to probe the interpretation in terms of
non-linearity of the steering magnets. Firstly, the magnetic fields produced by
two spare LHC corrector dipoles were precisely measured on a test bench, with
both standard magnetic cycles covering the full operating range, and test
cycles replaying the powering history during vdM scans at  from fill 6016
in 2017 <cit.>. These studies showed that, in the laboratory
environment, the magnetic fields are highly reproducible
(to better than $10^{-4}$),
but that there is a significant hysteresis effect depending on whether the
magnet current is being increased or decreased (and hence on the direction
of beam movement in a vdM scan).

Secondly, a series of test scans
were performed at  during the October 2021 LHC pilot run, as shown
in the lower part of Table <ref>. The DOROS BPM data were analysed
in the same way as for the 2018  scans, but keeping $M_1=1$ and
$M_{2,3}=0$, and without additional constraints from beamspot data.
Figures <ref>(a) and <ref>(b) show the residuals from the two
beams in a series
of $y$-plane scans in fill 7524. In this fill, the three bunches in each
beam were positioned around the LHC ring such that none collided in ATLAS,
eliminating beam–beam deflections.
Two consecutive parallel scans were performed, in which both beams were moved
from negative to positive displacement. These were followed by a separation
scan in which beam 2 was moved in the opposite direction (positive to negative)
to beam 1, as in a vdM scan, and then a further parallel scan. The offsets
between the DOROS BPM and nominal displacements were determined separately for
each scan, so that they are zero at the nominal zero displacement
just before each scan started. These residuals are not affected by beam–beam
displacements, and clearly show that the non-linearity at is highly reproducible
from scan to scan, with the opposite sign for beam 2 in the separation scan
where it moves in the opposite direction.

Residuals between the beam displacement at the
interaction point inferred from the calibrated DOROS BPM measurement and the
nominal beam displacement, during scans at  in the October 2021 LHC pilot
run. The left column shows B1Y and the right column B2Y; the results in the
$x$-plane are similar. The three rows show (a, b) a fill with no bunches
colliding at the ATLAS IP, (c, d) all three bunches colliding at ATLAS,
and (e, f) a `mixed' configuration with two out of four bunches colliding
at ATLAS. The beam–beam displacement was subtracted in plots (c–f), scaled by
the parameter  whose fitted values are given in the legends.

Figures <ref>(c) and <ref>(d) show results from fill 7525,
where all three bunches were colliding in ATLAS, and parallel and separation
scans were followed by scans where only one beam was moved and the other
remained  stationary (and is hence not plotted). The beam–beam displacement
was calculated
using the  value obtained from the separation scan S4ysep, and
subtracted from the residuals after being scaled by a fitted parameter ,
the same for all scans but different for the two beams.
The fitted values of  are within 10%
of unity, validating the modelling of the beam–beam displacement as seen by
the DOROS BPMs for the case where all bunches are colliding in ATLAS. The
residuals are again similar for all three scans for beam 1, and for the
separation and single-beam scans in beam 2 where it moves in the same direction.

Figures <ref>(e) and <ref>(f)  show results from fill 7516,
with four bunches
in each beam, and only two colliding in ATLAS. This was fitted in the same
way as for fill 7525, but the fitted  values are much smaller, 0.07
for beam 1 and 0.15 for beam 2. After subtracting this scaled beam–beam
displacement, the residuals are again
very reproducible between scans and similar to those in the other fills.
In such a fill, the DOROS BPMs see a mixture of colliding bunches that are
deviated by the beam–beam interaction, and non-colliding bunches that are not
deviated. The response of the DOROS electronics is known to depend both
on the time structure of the bunch pattern, and the relative intensity
of the bunches, and in this case the sensitivity to the beam–beam displacement
is reduced by much more than the $\bbds=0.5$ which would be expected from
the fraction of bunches that were colliding.

Qualitatively similar results were obtained in the $x$-plane, though with
a larger scatter of residuals. These studies confirm that the non-linearity
in the beam displacements appears to originate from hysteresis in the LHC
steering magnets, and is highly reproducible from scan to scan. The effect
of beam–beam deflections can be modelled by Eqs. (<ref>)
and (<ref>), after empirical scaling by the factor which must be determined from data.

To estimate the potential effect of magnetic non-linearity on the length scale
at , the data from each year's vdM and LSC scans were fitted in the same
way as at , fitting $M_{1,2,3}$ and  for each beam and plane
separately in each year. However, due to the smaller beam displacements compared
to the BPM and beamspot resolution, the results were less conclusive.
The residual distributions for some scan sessions showed shapes reminiscent
of the non-linearities seen at , and which could be eliminated by
adding $M_2$ and $M_3$ terms to the basic linear length scale. Others,
particularly 2016, showed no significant non-linearity, and some scans
in 2015 showed evidence of higher-order structure. In most datasets,
the fitted values of  are close to the naive expectation of
$\bbds\approx \nbun/\nbeam$, assuming the DOROS position readings correspond
to an unweighted average over all bunches.
The resulting values of $M_2$ and $M_3$
for each beam, plane and year are shown in Figure <ref>. Significant
$M_2$ and $M_3$ terms were fitted for many of the datasets, but with magnitudes
that typically do not exceed $1\,\mu$m. For a given beam and plane, the results
for different years are often very different, for both $M_2$ and $M_3$. This
may reflect the fact that the LHC orbit was set up independently in each year,
leading to potentially different corrector magnet currents and hence hysteresis
effects for the same scan patterns in different years.

The baseline fit was performed with a common value of  for all scans
for each individual beam and plane, and with the factor  of
Eq. (<ref>) set to $\sdd=2$, rather than $\sdd=1$ as used at
. This was required to get $\chi^2_\mathrm{doros}$ values close to one
per degree of freedom, suggesting that the DOROS BPM errors
estimated from the spread within each luminosity block do not fully capture
the measurement uncertainties. Alternative fits with $\sdd=1$, or
allowing  to vary per scan, were also considered. Since in the LSC scans
of 2015–2017, beam 2 was moved in the opposite direction
to that in the vdM scans, the baseline fits were performed with the signs
of $M_2$ and $M_3$ reversed in the LSC compared to the vdM scans, reflecting
the scan direction dependence suggested by Figures <ref>,
<ref> and <ref>. An alternative fit without these sign
reversals was also considered in these cases.

Summary of the non-linear length scale determination
for the  vdM+LSC datasets. The left and middle panels show the
quadratic $M_2$ and cubic $M_3$ terms fitted for each beam and plane from
the combined vdM and LSC data for each data-taking year, with error bars
indicating the statistical uncertainties. The right panel shows the linear
length scale product  for each year, together with its statistical
error (cyan band), the central value of the baseline non-linear fit
incorporating the $M_2$ and $M_3$ terms (black cross), and the total uncertainty
in  encompassing the central values of the baseline and alternative
non-linear fits (see text).

Given the lack of conclusive evidence for magnetic non-linearity at ,
the baseline  values were taken from the linear fits described in
Section <ref>, and the largest deviations from any of the
non-linear fits were considered to define a (symmetric) systematic uncertainty
added in quadrature to the statistical uncertainty from the linear fits.
This procedure assumes that any significant non-linearity can be
adequately described by a cubic parameterisation determined from
the combined fit to vdM and LSC scans, but does not attempt to correct it.
The resulting  values are shown in the rightmost panel of
Figure <ref>, together with the  values from the baseline
non-linear fits. The magnetic non-linearity uncertainty varies from 0.1% in
2016 to 0.6% in 2018, reflecting the fitted values of $M_3$ shown in
Figure <ref>.

§.§ Consistency checks

Since the value of  for a given luminosity algorithm should not depend
on beam parameters, the ability to determine it separately from each colliding
bunch pair and $x$–$y$ scan set allows the stability and reproducibility of the
calibrations to be checked. Figure <ref> shows the values measured for each bunch pair and scan in each vdM scan session. The
values are shown after all corrections have been applied, normalised to the
error-weighted mean over all bunch pairs and scans in each year.
Within one scan, the spread of  values over all bunch pairs
(after subtracting in quadrature the expected spread given the average
statistical error of the individual measurements) is a measure of
potential systematic bunch-to-bunch inconsistencies in the absolute luminosity
scale. The largest such value for each year was taken as the
`Bunch-by-bunch  consistency' uncertainty in Table <ref>.
These uncertainties
are at most 0.4% (in 2015), and zero in 2018, where the spread was consistent
with the expected bunch-to-bunch statistical fluctuations, which are larger than
in other years due to the use of a single LUCID PMT as reference algorithm.

Ratio of bunch-by-bunch visible cross-sections
to the weighted mean of  for all colliding bunch pairs
in all on-axis scans in the vdM scan sessions from each year of data-taking,
using the reference LUCID algorithm in each case. The vertical dashed lines show
the mean  for all bunches in each individual scan set.
The uncertainties are statistical, and are larger for 2018 due to the use
of the C12 single-PMT algorithm rather than a multi-PMT OR algorithm as used
in 2015–2017. The yellow bands show the combination of bunch-to-bunch and
scan-to-scan consistency uncertainties in each year, with the numerical
values indicated in the legends.

The differences in bunch-averaged  between scans give an indication
of the reproducibility of the calibration on a timescale of hours, or longer
in 2015 and 2016 where the scans were distributed over several LHC fills
(Table <ref>). The sampling-corrected standard deviation of these values
was used to define the `Scan-to-scan reproducibility' uncertainty in
Table <ref>, which is 0.3% or less in all years except in 2017, where
significant inconsistencies between scans are apparent, reflected in the larger
uncertainty of 0.7% that dominates the vdM calibration uncertainty for
that dataset. The average of the  values from all scans in one year
was finally used as the central value of the calibration result for that year.

In each year, the vdM calibration procedure was used to determine for all available LUCID and BCM algorithms. Since the acceptance of each
algorithm is different, the  values are also different, but
all algorithms should agree on the values of  and
hence on the specific luminosity  for each colliding bunch pair, given by

\begin{equation*}
\lspec = \frac{\lbun}{n_1 n_2} = \frac{f_\mathrm{r}}{2\pi \capsigx \capsigy}\ ,
\end{equation*}

and obtained from the vdM scan curves using the luminosity measurements
of each individual algorithm. The consistency of the different algorithms
was quantified by calculating the per-bunch-pair and per-scan relative deviation
of  for each algorithm compared to the average over all algorithms,
and then averaging this deviation over all bunch pairs and scans in each year.
Systematic deviations from this average are sensitive to problems with
individual algorithms
that may bias , such as detector non-linearity, short-term efficiency
drifts or background subtraction issues. The resulting relative deviations of
for all considered LUCID and BCM algorithms in each year are shown in
Figure <ref>, with the reference algorithm shown in red. The set of
available algorithms evolved during Run 2, as the LUCID configuration changed
and progressively more PMTs were equipped with bismuth calibration sources.
The BCM algorithms were not considered for the 2018 analysis, as they showed
significant instabilities during the vdM fill due to radiation-ageing effects
caused by the accumulated dose. The largest deviation from the average
in each year was used to determine the `Reference specific luminosity'
uncertainty in Table <ref>, which is at most 0.31%. The reference
algorithm has an   close to the
average of all algorithms in all years except 2018, where the single-PMT
algorithm C12 has an  0.31% higher than average. However, this algorithm
was chosen for the baseline luminosity measurement due to its availability and
stability throughout the year, as discussed in Section <ref>. The full
difference between the C12 and average  values is covered by the
2018 reference specific luminosity uncertainty.

Relative deviations of the specific luminosity
measured by each algorithm  from the average over all algorithms
$L_\mathrm{spec,avg}$, averaged over all colliding bunch pairs and scans
in the vdM session of each year. The uncertainties indicated by the error bars
are statistical, and the
symmetric uncertainty bands encompassing all algorithms in each year are shown
by the purple dotted lines. The reference algorithm for each year is indicated
by the red triangle.

The total uncertainties in the absolute vdM calibration are listed in
the `Subtotal vdM calibration' row of Table <ref>, and range
from 0.7% to 1.0% across the years of Run 2 data-taking.
The largest individual
uncertainties come from a variety of sources—non-factorisation effects in
2015 and 2016, scan-to-scan reproducibility in 2017 and magnetic non-linearity
in 2018. All other individual sources are smaller than 0.5% in all years.

§ CALIBRATION TRANSFER TO HIGH-LUMINOSITY RUNNING

The procedures described above provide the absolute calibration of the LUCID and
BCM luminosity measurements for low-$\mu$ data-taking with
a limited number of isolated bunches, where the instantaneous luminosities
are at least three orders of magnitude smaller than those typical
of normal physics running with bunch trains. The LUCID detector suffers
from significant non-linearity, and requires a downward correction
of $O(10\%)$ in the physics data-taking regime. This correction was derived
using track-counting luminosity measurements and validated with calorimeter
measurements, in particular those from the TileCal E-cell scintillators.
The calibration transfer procedures, and their
corresponding uncertainties, are discussed in detail below.

§.§ Comparison of luminosity measurements at high pileup

The natural luminosity decay during a long LHC physics fill, where the
luminosity, and hence bunch-averaged pileup , decrease by a factor of two
or three, provides a way to study the relative responses of the different
luminosity measurements as a function of . Figure <ref>
shows the ratios of track-counting luminosity to several other measured
luminosities in two long physics fills in 2017 and 2018, plotted as a function
of , which decreases steadily as the fills proceed. The ratios
of track-counting to TileCal D6-cell luminosities change by at
most 0.3% as the instantaneous luminosity decreases by a factor of three. The
EMEC measurements include a contribution from material activation, which
builds up over the first few hours of the fill to eventually contribute about
0.5% of the luminosity signal, leading to a somewhat larger decrease
in the track-counting to EMEC ratio of about 0.7% over the course of
these long fills. In contrast, the track-counting/LUCID ratios are
$O(10\%)$ smaller than unity at the start of physics fills, and exhibit a
strong dependence on  which is well-described by a linear function,
with different slopes in each year of data-taking. Although the track-counting
measurement was normalised to agree with LUCID in the head-on parts of the
corresponding vdM fills in each year, extrapolating the track-counting/LUCID
ratios to $\meanmu=0.5$ from the fits shown in Figure <ref> gives
ratios which differ from unity by $O(1\%)$, suggesting that the relative
response of the two detectors may also be affected by running with bunch
trains, and with a crossing angle between the colliding beams. The level
of agreement between the luminosities measured by track-counting, EMEC and
TileCal, and the strong disagreement between track-counting and LUCID,
suggest that residual $\mu$-dependencies in the track-counting, TileCal
and EMEC measurements are at the percent level or less, and that LUCID
overestimates the luminosity in physics conditions by $O(10\%)$, with a strong
pileup dependence.

Ratios of  (equivalent to ratios of
instantaneous luminosity)
measured by track-counting to that measured by LUCID (black points),
EMEC (red filled triangles) and TileCal D6 (blue open inverted triangles) as
a function of the mean number of interactions per bunch crossing measured by the latter algorithms in two long LHC physics fills in (a) 2017
and (b) 2018. The track-counting luminosity measurements are normalised to
those from LUCID in the head-on parts of the corresponding vdM runs
at low luminosity, and the calorimeter measurements are normalised to
give the same integrated luminosity as track-counting in the two physics
fills. The lines show linear fits through each set of points. The error bars
show the statistical uncertainties in the track-counting measurements. Time
progresses from right to left in these plots as  decreases through
the fill.

The relative dependence of the LUCID and track-counting luminosity measurements
was further studied by performing $\mu$-scans, where the luminosity was
varied between zero and the maximum achievable, by partially
separating the beams in the transverse plane at the ATLAS IP.
Since both LUCID and track-counting measure
$\mu$ for individual bunches, the performance as a function of the LHC bunch
pattern and of the position of individual bunches within a bunch train
could  also be investigated.
Figure <ref> shows results from a special fill (6194) recorded
in 2017, where the LHC filling pattern contained three isolated bunches,
two 48-bunch trains with 25 ns spacing and two 56-bunch 8b4e trains with a
repeating pattern of eight bunches separated by 25 ns followed by four empty
bunch-slots. During the $\mu$-scan, the ATLAS luminosity block boundaries
were synchronised to the steps in LHC beam separation, such that each LB
corresponds to a period with constant instantaneous luminosity at a different
$\mu$ value. The LBs where the luminosity was varying as the beam separation
was changed were discarded. Figure <ref>(a) shows the
ratio of $\mu$ from track-counting to that from LUCID averaged over the
three isolated bunches, as a function of $\mu_\mathrm{LUCID}$.
There is a significant negative slope (indicating that LUCID overestimates
the luminosity compared to track-counting at high pileup), but the slope
is several times smaller than those seen in Figure <ref>.
Figure <ref>(b) shows the same ratio averaged over all the bunches
in the 25 ns and 8b4e trains separately. Here, the slopes are larger,
demonstrating a larger $\mu$-dependence in train bunches, which is slightly
stronger for 25 ns than 8b4e bunch trains. Deviations from the linear slope
are also visible at the lower and upper ends of the $\mu$ range. Due to the
spread in the bunch intensities and emittances produced by the LHC injector
chain, the $\mu$ range sampled for 8b4e trains is slightly larger than
that for 25 ns trains, and the isolated bunches extend almost to $\mu=100$,
much larger values than those encountered in normal LHC fills during Run 2.

Mean values of the bunch-by-bunch ratio of
$\mu$ measured by track-counting and LUCID as a function of the $\mu$
measured by LUCID from the $\mu$-scan in LHC fill 6194 with a mixture of
isolated bunches, 25 ns and 8b4e trains. The ratios are shown separately for
(a) isolated bunches and (b) the two types of bunch trains. The lines show
linear fits through each set of points. The error bars show the uncertainties
in the track-counting measurements.

The $\mu$-ratios between track-counting and LUCID were also analysed as a
function of bunch position $b$ within the train, fitting the ratio $R^b=\mu^b_\mathrm{Tracks}/\mu^b_\mathrm{LUCID}$ as a function of $\mu_\mathrm{LUCID}$ to a
linear function with intercept $p_0^b$ and slope $p_1^b$.
The resulting fitted parameters are shown separately for the 25 ns and 8b4e
bunches in Figure <ref>. Whilst the intercepts $p_0$ are similar in
isolated bunches and both types of train bunches, and do not depend significantly
on train position, the slopes $p_1$ show significant structure. The first bunch
of a 25 ns train has a similar $p_1$ (and hence $\mu$-dependence) to that
of isolated bunches. The subsequent bunches show increasingly negative
$p_1$ (and larger $\mu$-dependence), reaching a plateau about six bunches
into the train, after which there is a slight recovery to a constant value
through the bulk of the train. The 8b4e trains show a similar initial behaviour,
but the four bunch-slot gap is enough for the ratio to `recover', and the
first bunch of the next eight-bunch subtrain again behaves similarly to
an isolated bunch. This structure is consistent with the slightly smaller
$\mu$-dependence for 8b4e compared to 25 ns trains seen in
Figure <ref>.

Fitted intercept $p_0$ (upper plots) and slope
parameter $p_1$ (lower plots) for the track-counting/LUCID $\mu$ ratios in
LHC fill 6194 as a function of position within the bunch train, counting the
first bunch in the train as position zero. The left plots show the results for
the 48-bunch 25 ns trains, and the right plots those for the 56-bunch 8b4e
trains with eight colliding bunches separated by 25 ns followed by a four
bunch-slot gap. The right-most points in each plot show the corresponding values
for isolated bunches. The error bars show the statistical uncertainties from
the track-counting measurements.

Since the calorimeter measurements cannot resolve the luminosity of individual
bunches, the bunch-by-bunch stability of track-counting measurements
cannot be validated except by `internal' comparisons of different track-counting
working points (see Section <ref> below). However, given the small
relative $\mu$-dependence between the bunch-integrated track-counting
measurements and the EMEC and TileCal calorimeter measurements
shown in Figure <ref>, it is reasonable to attribute
the bulk of the effects seen in Figures <ref>
and <ref> to LUCID. A likely contributor is `signal migration'
<cit.>, the overlap of several below-threshold signals in the same LUCID
PMT from different
$pp$ interactions, which combine to produce an above-threshold signal
which gets counted as a hit, causing deviations from the Poisson assumption in
Eq. (<ref>). These effects become more likely for higher $\mu$ values,
leading to an overestimate of the luminosity at high pileup. The structures
seen in Figure <ref> also suggest that particles from previous
bunch crossings can contribute to this signal migration, making the overestimate
worse for bunches further from the start of the train.

§.§ Performance of track-counting algorithms

The performance of the track-counting luminosity measurements was studied
using the standard ATLAS detector simulation <cit.> based
on Geant4 <cit.>. A sample of events containing multiple overlaid
inelastic $pp$ collisions was generated with Pythia 8.186
<cit.> and the A3 set of tuned parameter values
with a flat $\mu$ profile in the range $0<\mu<100$.
The sample included the simulation
of bunch trains, with 53 trains each containing 48 colliding-bunch pairs
with 25 ns spacing, simulating some of the effects of in-time and out-of-time
pileup. These events were processed using the same dedicated track
reconstruction settings as used for the track-counting event stream in
real data, only making use of information from the pixel and SCT detectors.
Figure <ref>(a) shows the efficiency to reconstruct a track
and select it using each of the working points (normalised to the
true number of tracks satisfying the  and $|\eta|$ requirements for
each working point  shown in Table <ref>), as a function of $\mu$.
For the baseline selection A, the efficiency decreases by only about 0.3%
between $\mu=0$ and $\mu=100$. The efficiencies for the other two working
points decrease faster, reflecting the more stringent hit requirements
(see Table <ref>).
Figure <ref>(b) shows the fraction of fake tracks
(tracks which cannot be matched to a generator-level track with a hit-based
matching probability $P_\mathrm{match}>0.5$ as defined in
Ref. <cit.>), which is around 0.1% at $\mu=50$ for
selection A,  rising non-linearly to about 1% at $\mu=90$ as the
detector occupancy increases. Here, the alternative selections show
smaller fake-track rates, again reflecting the more stringent hit requirements.
The simulation includes some of the effects which lead to a loss of performance
at high pileup, such as the `01X' hit pattern definition in the SCT, which
requires that there be no hit on a strip in the immediately preceding
bunch-crossing for a valid hit to be read out. However, it does not include
an inefficiency in the pixel detector due to time-over-threshold effects,
where a pixel which is hit again whilst the decaying signal from an initial hit
is still over the readout threshold will not record the second hit. This
leads to an inefficiency which may last for tens of bunch-crossings, and
particularly affects selection B, which rejects
tracks with a missing pixel hit where one is expected.

Performance of the different track-selection working
points for the track-counting luminosity measurement in simulation as a
function of $\mu$, showing (a) combined track reconstruction and
selection efficiency, and (b) fake-track probability.

The efficiencies of the track-selection working points were also studied using
$Z\rightarrow\mu\mu$ events in data, where both muons were reconstructed
as isolated `combined' muons with $\pt>20$ , consisting of matched tracks
in the ATLAS inner detector
and muon spectrometer <cit.>. The fraction of the inner detector
tracks associated with these muons that pass the track-counting selections,
which only rely on criteria related to the inner detector, gives
a measure of the inner detector track selection efficiency relative to the
inclusive initial selection. The latter just requires a very loose reconstructed
track passing $\pt$ and $|\eta|$ requirements, together with minimal
requirements on the number of inner detector hits to ensure the track can be
reconstructed <cit.>.  Figure <ref>(a)
shows the resulting efficiencies measured in 2018 data for the different
working points. The efficiency for selection A
is reasonably stable with $\mu$, decreasing by about 0.4% between
$\mu=0$ and $\mu=60$, a decrease which is however larger than that seen in
simulation. The efficiencies of the other selections show much stronger
dependence on $\mu$, especially for selection B. Figure <ref>(b)
shows the efficiencies measured in part of the 2017 8b4e data-taking period
as a function of
the bunch position along the train. Again, the selection A working point
is  rather stable, decreasing by about 0.2% along the eight-bunch subtrains,
whilst selection B shows $O(1\%)$ effects.
Although these efficiencies are measured for muons with $\pt>20$ ,
simulation studies showed that they also capture the main effects for the
track-counting selections, which are dominated by hadrons with lower .

Track-selection efficiency for the
track-counting luminosity measurement determined using $Z\rightarrow\mu\mu$
events in data: (a) Efficiencies for the different selections as a
function of  in 2018 data; (b) Efficiencies
for data recorded during part of the 8b4e collision period as a function of the
position of the bunch within the 8b4e train, numbering the first bunch
as position zero.

Figure <ref> shows the selection efficiencies measured
using $Z\rightarrow\mu\mu$ events  as a function of
time over the entire 2018 dataset. Selections A and C are very
stable, changing by less than 0.2% during the course of the year.
Selection B is much less stable, with about a 1% lower efficiency in the
first few weeks. This was traced to an incorrect masking of inactive pixel
modules in the track reconstruction, leading to more tracks with unexpected
pixel holes. Since this early data also had atypically
high  values, this effect also causes the decrease in efficiency
for $\meanmu>50$ visible for selection B in Figure <ref>(a).
Similar levels of stability for the selection A working point
were seen for the datasets from other data-taking years within Run 2. Given
these observations, no attempt was made to use the $Z$-based efficiency
measurements to correct the track-counting
luminosity as a function of time during each year.

Track-selection efficiency for the track-counting
luminosity measurement determined using $Z\rightarrow\mu\mu$ events as a
function  of time within the entire 2018 high-luminosity physics dataset.
The efficiencies of the different track selection  working points are shown

These results using both simulation and real collision data demonstrate that
the track-counting luminosity measurement with the selection A working point is
much more robust than the LUCID luminosity measurements against variations of
$\mu$ and bunch position. The track-counting measurement also shows
excellent stability in time, and can thus be used as a reliable relative
reference to calibrate other luminosity algorithms, once its absolute
calibration has been determined by normalisation to LUCID in the vdM run.

§.§ LUCID correction strategy

Exploiting the excellent stability of the track-counting luminosity measurement,
the non-linearity in LUCID was corrected as a function of  using one
or more long high-luminosity reference fills in each year of data-taking. Each
reference fill was used to derive a correction of the form

\begin{equation}\label{e:mucorr}
\mmucorr = p_0 \mmuuncorr + p_1 (\mmuuncorr)^2\ ,
\end{equation}

where  is the uncorrected and  the corrected LUCID value. The parameters $p_0$ and $p_1$ were obtained from a linear fit to
the ratio of  values measured by track-counting and LUCID,
$R=\langle\mu_\mathrm{Tracks}\rangle/\langle\mu_\mathrm{LUCID}\rangle$, as a
function of , as shown in Figure <ref>. The track-counting
luminosity was first normalised to the absolute luminosity measured by LUCID
in parts of the vdM fill with stable head-on collisions at ATLAS, ensuring
that $R=1$ at low pileup ($\mu\approx 0.5$) with isolated bunches.
This correction also absorbs the $O(1\%)$ change in the LUCID calibration caused
by the non-zero LHC beam crossing angle used in physics runs, and the
$O(0.2\%)$ afterglow contribution to the LUCID luminosity signal
in bunch train running. In principle,
Eq. (<ref>) could also be written in terms of the bunch-by-bunch
$\mu$, correcting the LUCID measurement as a function of the instantaneous
luminosity in each individual bunch and potentially compensating for the
bunch-to-bunch variations seen in Figure <ref>. However, since
ATLAS physics analyses make use of the data from all LHC bunches, an
average correction based on  is sufficient, and a more complex
bunch-position-dependent correction scheme was not used.

The correction of Eq. (<ref>) ensures that the LUCID luminosity agrees
well with track-counting throughout the reference fill. If the LUCID
response has no time dependence, a single reference fill and correction
would be adequate for a whole year of data-taking. The regular PMT high-voltage
adjustment based on the LUCID $^{207}$Bi calibration signals ensures that
this is approximately the case, but comparisons with track-counting show
some residual variations, particularly during the first few weeks of
data-taking in each year, when the number of colliding bunches and
instantaneous luminosity increased during the LHC intensity ramp-up phase.
Better agreement with track-counting throughout the year was therefore
obtained by dividing each year into several correction periods, or epochs,
with a long LHC fill representative of each epoch being used as a reference run.
The change from 25 ns to 8b4e bunch trains in 2017 also required separate
epochs for the two periods (the 8b4e epoch also includes a
period of unstable 25 ns running with a reduced number
of bunches and many short fills immediately preceding the change to 8b4e).
One epoch in 2015, three epochs in each of 2016
and 2017, and two in 2018 were used in order to achieve good agreement
between LUCID and track-counting throughout each year. Figure <ref>
shows the resulting level of agreement between track-counting
and LUCID for each data-taking run (normally equivalent to one LHC fill),
in terms of the relative differences of the per-run integrated luminosities
measured by the two algorithms. The differences are plotted as a function
of cumulative integrated luminosity through the year normalised to the per-year
total, giving a fraction which increases
monotonically from zero to one as a function of time within the year.
In all years, the run-integrated LUCID and
track-counting measurements typically agree to better then 0.5%, and at least
about 70% of the data in each year is covered by a single correction epoch.