percent), indicating that the effect of e.g. emittance growth and beam drifts
during the short intervals is negligible. In the following, sources of
systematic uncertainties and their effect on the measurement precision will be
described.

Figure 30: Evolution of normalized cross-section and related values for a
colliding bunch pair for BCID 1335 of (left) fill 2855 and (right) of fill
3311. The cross-section values are normalized to their average while other
parameters are normalized to the first data point. Fill 2855 lasted more than
eight hours with a beam intensity product decrease of about 10%. Fill 3311
lasted about 4h30 with a beam intensity product decrease of less than 2%; in
this fill the luminosity reduction is mostly caused by emittance growth.

The impact of the beam-beam interaction vertex resolution on the cross-section
depends on the transverse size of the luminous region. Comparing the results
obtained with different ratios of the resolution to this transverse size
permits evaluating the corresponding systematic uncertainty. The measurement
of the width of the luminous region can be biased by e.g. a misalignment in
the VELO sensors, which would correlate the position of the sub-vertices and
broaden the beam without being detected by the resolution measurement.
Considering head-on beams by setting $\Delta x=\Delta y=0$ in Eq. (17), one
sees that $\Omega\propto({\Sigma_{x}\Sigma_{y}})^{-1}$. Assuming similar size
for both beams in each transverse coordinate, one can derive from the
formalism for pure Gaussian beams that the quantity

$R=\frac{2\sigma^{2}_{\text{res},lx}}{4\sigma^{\prime
2}_{lx}+\sigma^{2}_{z}\sin^{2}\phi_{x}}+\frac{\sigma^{2}_{\text{res},ly}}{2\sigma^{\prime
2}_{ly}}$ (35)

is representative of the importance of the beam-beam resolution in the cross-
section measurement. Here, the $\sigma^{\prime}_{lm}$ represent the measured
values of $\sigma_{lm}$ and $\sigma_{\text{res},lm}$ the beam-beam resolutions
($m=x,\,y$). A value of $R=0$ means that the resolution is negligible compared
to the beam size. Cross-section measurements with different $R$ values are
obtained by using different cuts on the vertex track multiplicity and by using
data acquired with a $\beta^{*}$ value of 3 and 10$\rm\,m$. This procedure
gives four sets of results with different values of $R$.

Figure 31: Illustration of the evaluation of the systematic uncertainty due to
the beam-beam resolution. Measured visible cross-section values for different
event samples are shown as a function of $R$. The $R$ parameter is calculated
with Eq. (35). Each data point is an average of all measurements from a fill.
The error bars are the RMS of the cross-section and $R$ values per fill. Plain
markers are measurements performed with $\beta^{*}=10\rm\,m$ (6 fills) and
open markers are performed with $\beta^{*}=3\rm\,m$ (2 fills). The samples
with $R\approx 0.5$ are selected to enhance the effect of the resolution. The
dashed line visualizes the determination of the systematic uncertainty as
described in the text.

The results for the six fills with $\beta^{*}=10\rm\,m$ and two fills with
$\beta^{*}=3\rm\,m$ are combined in Fig. 31. Measurements performed with the
larger $\beta^{*}$ and the best resolution (high track multiplicity cut)
provide the smallest $R$ value, while a worse resolution combined with the
smaller $\beta^{*}$ results in a larger $R$ value. The clear correlation
between the cross-section and the $R$ value visible in Fig. 31 is an
indication that the resolution is not perfectly understood. The cross-section
obtained for similar $R$ values and different $\beta^{*}$ are similar. This
shows that the difference between the measurements at $\beta^{*}=3\rm\,m$ and
$\beta^{*}=10\rm\,m$ can be attributed to the resolution description. The
leftmost group of data points are the results presented in Fig. 29. Those
results are obtained with the high track multiplicity cut and with a
$\beta^{*}=10\rm\,m$ beam optics. This combination provides the best
measurement conditions and those measurements, called here $\sigma_{c}$ are
used as central value for the final results. The cross-section $\sigma_{e}$
obtained by extrapolating the cross-section to $R=0$ based on the
$\beta^{*}=10\rm\,m$ fills with the low and high track multiplicity cuts (blue
and magenta leftmost measurement groups) is used to evaluate the uncertainty
due to the beam-beam resolution. The difference of
$\Delta\sigma=\sigma_{c}-\sigma_{e}=0.93\%$ between the central value
$\sigma_{c}$ and the extrapolated value $\sigma_{e}$ is taken as systematic
uncertainty. This is the largest single contribution to the uncertainty of the
cross-section result.

As discussed in Sec. 6.2, a set of correction factors to the beam-gas
interaction vertex resolution have been determined to reach consistent beam
width measurements at different $z$ positions. The necessity to include
correction factors is an indication of additional systematic uncertainties.
The overlap integrals have also been evaluated without the resolution
correction factors. The full difference between the results with and without
correction factors amounts to 0.55%, which is used as the corresponding
systematic uncertainty.

A misalignment of the VELO detector can correlate the positions of the sub-
vertices and degrade the vertex resolution. This effect is included in the
systematic uncertainties assigned to the resolution as described above.
However, detector misalignment can also affect the crossing-angle measurement.
Various versions of the detector alignment are produced, all of which provide
acceptable and comparable results in the overall alignment quality. The
different alignment versions arise from introducing different sets of
constraints to satisfy internal consistency checks. The same data set is
reconstructed with all alignment versions and measurements of the half
crossing angle are performed. In the $x$ direction, differences of the order
of 10 µrad are found, an order of magnitude larger than the statistical
uncertainty. The crossing angle uncertainty in the $y$ axis is about 3 µrad
and has a negligible impact on the luminosity. The luminosity uncertainty from
the crossing angle correction depends on the bunch width and length and is
different for each bunch pair. The full range of 0.45% in the various measured
cross-sections for different alignment versions is taken as systematic
uncertainty related to this source. The different alignment versions have
negligible impact on the bunch shape measurements.

The bunch shape model defined in Sec. 5.3 can describe all observed bunch
shapes with a $\chi^{2}/$ndf close to one. Nevertheless, the accuracy of the
fit procedure as well as the capacity to describe different shapes is
verified. The simulation (Sec. 6.4) is used to generate datasets of vertex
distributions with different input parameters in order to test the capability
to reproduce the input values by the fit procedure. In addition, the
systematic uncertainty arising from fitting a double Gaussian model to
different simulated shapes is estimated. Single, double or triple Gaussian
shapes, or a “Super Gaussian shape”666A generalized Gaussian shape with a
power parameter different from 2. [43], are tried.

A given set of input parameters is used to generate a sample of statistically
independent datasets, each of which is analysed using the same algorithms as
for real data. All shapes give a $\chi^{2}/$ndf close to unity and can be
considered to be well described by the fit model. The results of the
measurements of the simulated data are compared to the input parameters on the
basis of the value of the overlap integral. The difference in the results
indicates a 0.5% systematic uncertainty due to the fit model and accuracy of
the fitting procedure. The double Gaussian fit model used to describe the
transverse bunch shapes does not allow a description of a possible third
component. For example, a fraction of the protons measured in a bunch could be
present in a wider Gaussian shape that would not be measured with a double
Gaussian function. In this case, the tails of the measured distributions would
have a larger population fraction than expected. The fraction of vertices in
the tails beyond the double-sided 99 percentile predicted by the fit, is
checked for all measurements with $\beta^{*}=10$ m and for the simulated
datasets. The tail population is about 2% for the single beams and about 1.5%
for the luminous region (while 1% is expected). This tail population, however,
is also observed in the simulated datasets, indicating that the higher tail
population is a result of the fitting procedure. Therefore, the corresponding
bias is already included in the fit model uncertainty given above, and no
additional systematic uncertainty is assigned.

Measurements of cross-sections for all colliding bunch pairs in the fills of
2012 with $\beta^{*}=10$ m are shown in Fig. 32. The measurements use the
_Vertex_ observable as reference and have an RMS spread of 0.54%. Cross-
section results are also shown on a fill-by-fill basis in Fig. 32; the
indicated error bar is the RMS of all measurements of the corresponding fill.
Since the statistical uncertainty per measurement is significantly less than
0.5%, the observed spread of 0.54% is due to the combination of statistical
fluctuations and additional systematic effects. The full RMS is taken as
systematic uncertainty on the cross-section. This uncertainty covers
uncorrelated bunch-by-bunch uncertainties such as the shape description, which
is influenced by the fit model and detector resolutions, or uncertainties in
the bunch population measurements.

Figure 32: Overview of cross-section measurements for head-on periods in fills
with $\beta^{*}=10\rm\,m$ at $\sqrt{s}=8\rm\,TeV$ with (left) the histogram
combining all data from Fig. 29 and (right) the central value of cross-
sections averaged per fill. The dashed vertical line indicates the median
value of all measurements. The error bars indicate the standard deviation of
all measurements from a fill. The statistical uncertainty of each average is
smaller than the marker size.

Some structure remains visible in the measurement shown in Fig. 29, pointing
towards additional systematic uncertainties or a less than perfect bunch shape
description. Correlations between the cross-sections and the major variables
entering into the cross-section measurement (interaction rate, bunch
population product, overlap integral, luminous region $z$ position, crossing
angle corrections) have been checked and are found to be negligible.

As described in Sec. 6.3, the convolved bunch length
$\sigma_{z1}^{2}+\sigma_{z2}^{2}$ and bunch crossing position $z_{\rm rf}$ are
measured with a fit to the longitudinal distribution of the luminous region.
For each colliding bunch pair measurement, all vertices in the range
$|z|<250\rm\,mm$ are selected, regardless of the track directions. This
selection reduces the effect of the $z$ dependence in the VELO region and
limits the distortion of the luminous region shape. On the other hand, some
beam-gas interaction vertices can be included in the distribution. An
additional measurement of $\sigma_{z1}^{2}+\sigma_{z2}^{2}$ and $z_{\rm rf}$
is performed by requiring that vertices contain at least two forward and two
backward tracks to exclude beam-gas interactions. This requirement distorts
its longitudinal shape as backward tracks with an origin $z\gtrsim-95\rm\,mm$
are not detected by the VELO. The reconstruction efficiency $\epsilon_{\rm
vtx}$ obtained from simulation and corresponding to this requirement is shown
in Fig. 33 and is used to correct the raw distribution. An example of a fit to
the longitudinal distribution of the luminous region with this requirement is
also shown in the figure for one bunch-pair measurement. All cross-section
measurements have been analysed with both track requirements. The difference
between the results obtained with the two methods is 0.04% and is taken as
systematic uncertainty related to the reconstruction efficiency.

Figure 33: Determination of the bunch length of a colliding bunch pair with
(left) the vertex reconstruction efficiency as function of $z$ and (right) the
fit to the longitudinal vertex distribution after efficiency correction. The
statistical uncertainty on the corrected points is dominated by the limited
amount of simulated data. This component of the uncertainty is not shown in
the right-hand plot. The data are selected requiring vertices with at least
two forward and two backward tracks. The efficiency is normalized to the value
at $z=0$ to keep similar amplitudes between the raw and corrected data; the
absolute scale of the correction does not change the fit results. This example
displays data for BCID 1335 in fill 2855. The raw data (black dots) show the
distortion resulting from the requirement of having at least two tracks in
both directions. Figure 34: Measurement of convolved bunch length
$\sigma_{z1}^{2}+\sigma_{z2}^{2}$ with displaced beam during fills 2852 and
2853. The fills are indicated in the plot and separated by a dashed vertical
line. The measurements are sorted by time and BCID. About 3.5 hours separate
the first head-on period in fill 2853 from the period with offset beams
explaining the higher values of the bunch length observed during that period.
The next head-on period follows the period with offset beams without time
interval.

Two dedicated BGI measurements were performed with beams displaced vertically
with repect to each other by 180${\,\micro\rm m}$ in fills 2852 and 2853 with
a $\beta^{*}=10\rm\,m$ optics and were preceded and followed by periods with
no displacement. The beam offset reduced the overlap integral by a factor of
about four. The convolved bunch length $\sigma_{z1}^{2}+\sigma_{z2}^{2}$
measured during those fills is shown in Fig. 34. Periods where the beams were
offset show the same $\sigma_{z1}^{2}+\sigma_{z2}^{2}$ values as when the
beams were head-on. With a relative displacement of 180${\,\micro\rm m}$, the
luminosity is dominated by the interaction of the wide beam components of both
beams. On the other hand, the luminosity is dominated by the interaction of
the narrow beam components when the beams collide head-on. The equality of the
convolved bunch length values for head-on and offset collisions is an
indication that the wide and narrow bunch components share the same length,
justifying the assumption that the $z$ coordinate is factorizable. The
observed difference of 0.05% in the cross-section for the periods with offset
beams compared to the head-on beams is taken as systematic uncertainty related
to the convolved bunch length $\sigma_{z1}^{2}+\sigma_{z2}^{2}$ measurement.

The beam-gas interaction rate is proportional to the residual gas pressure at
the interaction point. The BGI method measures the beam shapes with beam-gas
interaction vertices, and therefore assumes that the pressure is uniform in
the transverse directions. The relative error induced by a pressure gradient
is estimated by evaluating the effect of a distortion on the overlap integral
considering a constant pressure along the $y$ axis and a pressure gradient
along the $x$ axis at the experimentally determined limit. A measurement was
performed in 2010 during fill 1422 to verify the homogeneity of the pressure
in the $x$ direction by displacing the beams by 0.3$\rm\,mm$ and has been used
in the luminosity measurement for 2010 [10]. The relative uncertainty on
$\Omega$ introduced by the pressure gradient is at most 0.03%.

The VELO transverse dimensions, which fix the absolute scale of the vertex
transverse distribution measurements, were checked in the laboratory at
different temperatures on individual silicon sensors. The relative uncertainty
on the cross-section determination due to the transverse scale uncertainty is
estimated to be at most 0.05%.

The FBCT response is approximatively linear with respect to bunch intensities.
However, small deviations from linearity lead to a non zero offset when
extrapolating to zero bunch intensity. This offset can be inferred from the
combination of all cross-sections measured during a fill, taking advantage of
the fact that the cross-section is independent of the choice of the bunch
pair. An analysis of the FBCT non-linearity is performed as outlined in Ref.
[26]. A fit is performed with the offsets and the improved cross-section value
as free parameters. As expected, the $\chi^{2}/$ndf values are improved by
this procedure. All cross-section results obtained after applying the FBCT
offset corrections for fills with $\beta^{*}=10\rm\,m$ and
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ give a distribution with an RMS of
0.48% instead of 0.54% without the correction. The central value is changed by
0.04%. This small deviation is not applied to the final result, but is taken
as systematic uncertainty related to a potential FBCT offset. The systematic
uncertainties introduced by the bunch population determination are described
in Sec. 3.

A difference of 0.2% is observed between the background-subtracted interaction
rate measurement with the restricted _Vertex_ observable and the corresponding
unrestricted observable. This difference is attributed to the background
subtraction and assigned as systematic uncertainty. The reference cross-
section used for physics data taking is based on the _Track_ observable, which
is affected by beam-gas interaction background when the neon gas injection
system is used for the BGI calibration data taking. To transport the visible
cross-section based on the _Vertex_ observable to that based on the _Track_
observable, their ratio is measured in periods without gas injection. A
variation of 0.2% is observed, which is taken as systematic uncertainty. Using
the relation between the _Vertex_ and _Track_ observable of
$\mu_{\text{Track}}/\mu_{\text{Vertex}}=1.106$, the final calibration result
is $\sigma_{\text{Track}}=60.62\pm 0.87\rm\,mb$. A summary of all
uncertainties is provided in Table 7. The values shown for the other energies
will be discussed below.

Table 7:  Relative systematic uncertainties on the reference cross-section for the BGI calibrations at $\sqrt{s}$ values of 8, 7 and 2.76$\mathrm{\,Te\kern-0.90005ptV}$ (in %). The uncertainties are divided into groups affecting the measurement of the overlap integral (Sec. 6), the bunch population product (Sec. 3) and the rate measurement (Sec. 4).  Source of uncertainty | 8$\mathrm{\,Te\kern-1.00006ptV}$ | 7$\mathrm{\,Te\kern-1.00006ptV}$ | 2.76$\mathrm{\,Te\kern-1.00006ptV}$
---|---|---|---
Overlap integral measurement
Beam-beam resolution | 0.93 | 1.00 | 0.40
Beam-gas resolution | 0.55 | 2.80 | 1.31
Detector alignment | 0.45 | 0.30 | 0.90
Fit model | 0.50 | 1.50 | 0.50
Measurement spread | 0.54 | 1.00 | 1.30
Reconstruction efficiency | 0.04 | 0.20 | 0.17
Bunch length and $z_{\rm rf}$ | 0.05 | 0.10 | 0.10
Pressure gradient | 0.03 | 0.03 | 0.03
VELO transverse scale | 0.05 | 0.05 | 0.05
Statistical uncertainty | 0.01 | 0.25 | 0.03
Bunch population uncertainties
FBCT non-linearity | 0.04 | 0.05 | 0.05
DCCT population product | 0.22 | 0.24 | 0.28
Ghost charge | 0.02 | 0.10 | 0.07
Satellite charge | 0.06 | 0.25 | 0.08
Rate measurement
Background subtraction | 0.20 | 0.05 | 0.20
Ratio of observable _Track_ to _Vertex_ | 0.20 | n.a. | 0.20
Final $\delta\sigma_{\text{Track}}/\sigma_{\text{Track}}$ | 1.43 | 3.50 | 2.20

###  6.7 Beam-gas imaging results at lower energies

The analysis presented in this paper focuses on the calibration of the
reference cross-section at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ ($pp$)
since it gives the most precise results. Comparable results are obtained with
the BGI method for the 2013 $pp$ calibration at
$\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$ and the 2011 $pp$ calibration at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$.

The $\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$ calibration was performed in
three dedicated fills (3555, 3562 and 3563). Trigger conditions were identical
to those used in July and November 2012 and the neon gas injection system was
active. The luminosity measurement and evaluation of systematic uncertainties
follows the same procedure as with $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$
data. Cross-section results for $\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$
are shown in Fig. 35. The RMS of the measurements is 1.3%.

Figure 35: Cross-section results with head-on beams with $\beta^{*}=10$ m at
$\sqrt{s}=2.76$ TeV for the _Vertex_ observable, with (left) the individual
measurements and (right) the histogram of the values. The individual
measurements are made per colliding bunch pair and 20 minutes time integration
and are sorted by time and BCID. The different fills are separated by a dashed
vertical line in the figure. Two dotted horizontal lines show the $\pm 1\%$
deviation from the central value. The measurement spread has a 1.3% RMS. The
median value is indicated by a dashed vertical line in the right-hand plot.

A half crossing angle of 885 µrad was chosen to avoid collisions between
satellite and main bunches. Differences observed in the data as compared to
2012 data are discussed below. The beams have a double Gaussian shape and are
significantly non-factorizable ($f_{j}$ is close to zero). However, the
factorizability is more difficult to measure than with
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ data and the uncertainties on the
$f_{j}$ parameter are larger. The beam-beam resolution has a small impact on
the cross-section compared to 2012 (uncertainty of 0.40%), owing to the
comparatively large transverse bunch size. On the other hand, the uncertainty
related to the beam-gas vertex position resolution has significantly larger
impact than in 2012, due to the lower number of vertices in the luminous
region. The cross-section difference measured with and without all beam-gas
resolution correction factors amounts to 1.31%. The uncertainty related to the
detector alignment of 0.9% is estimated at twice the value obtained in 2012
because the crossing angle correction is about twice as large. The convolved
bunch length $\sigma_{z1}^{2}+\sigma_{z2}^{2}$ measurement plays a more
important role compared to the $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ data
due to the larger crossing angle correction. An uncertainty of 0.1% is
assigned to the determination of $\sigma_{z1}^{2}+\sigma_{z2}^{2}$ and $z_{\rm
rf}$ and 0.17% to the reconstruction efficiency. The FBCT offset fit changes
the cross-section by 0.05% and reduces the overall RMS to 1.1%. The systematic
uncertainty related to the ghost charge amounts to 0.07%. Uncertainties for
the observable background subtraction and fit model are taken from the 2012
measurements. The reference cross-section for the _Vertex_ observable is
$46.4\pm 1.0\rm\,mb$. A summary of all uncertainties is provided in Table 7.
As for the $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ data, the reference
cross-section used for physics data taking is based on the _Track_ observable.
Using the relation between the _Vertex_ and _Track_ observable of
$\mu_{\text{Track}}/\mu_{\text{Vertex}}=1.135$, the final calibration result
is $\sigma_{\text{Track}}=52.7\pm 1.2\rm\,mb$.

In the 2011 $pp$ calibration, the main differences with the situation in 2012
are the absence of the neon gas-injection system and a different trigger
configuration. Therefore, the beam-gas interaction rate is a factor 20 lower
than in 2012. To partly compensate for the lower rate, measurements are
performed in one-hour periods, potentially introducing effects of emittance
growth and beam drifts. In addition, data are available in one dedicated
calibration fill only. Cross-sections results for
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ are shown in Fig. 36. The RMS of the
measurements is 2.0%.

Figure 36: Cross-section results at $\sqrt{s}=7$ TeV in 2011 for the _Track_
observable, with (left) the individual measurements and (right) the histogram
of the values. The individual measurements are made per colliding bunch pair,
for a one hour integration time, and are sorted by time and BCID. The
different fills are separated by a dashed vertical line in the figure. Two
dotted horizontal lines show the $\pm 1\%$ deviation from the central value.
The measurement spread has a 2.0% RMS. The median value is indicated by a
dashed vertical line in the right-hand plot.

Collisions between satellite and main bunches were observed at $z=\pm 375$ mm
due to the low half crossing angle of 270 µrad. Therefore, beam-gas vertices
in the region $-550$ mm $<z<550$ mm are discarded to exclude beam-beam
vertices in the single beam selection. This requirement, however, reduces the
number of vertices available for the single beam measurement. Furthermore, the
remaining vertices are measured with a worse resolution leading to a bigger
impact on the beam size measurement. The limited amount of beam-gas vertices
reduces the accuracy of the resolution determination and the correction
factors described in Sec. 6.2 cannot be determined. An uncertainty of 10% is
assumed for the beam-gas vertex resolution leading to an uncertainty of 2.8%
on the cross-section calibration.

The ratio of the resolution width to the beam size is smaller in 2011,
potentially reducing the uncertainty resulting from the beam-beam resolution.
However, the same uncertainty as for $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$
is conservatively assumed for the impact of the beam-beam vertex resolution.
Given the smaller crossing angle, the uncertainty related to the detector
alignment effects is estimated at 0.3%. The beam factorizablility is more
difficult to measure in 2011 due to the lower number of vertices. The beams
are mostly non-factorizable ($f_{j}$ is typically close to zero).

Owing to the low beam-gas induced background, the reference cross-section can
be based on the _Track_ observable directly without using the ratio _Vertex_
to _Track_. A global satellite fraction correction of 0.78% is applied to the
result shown in Fig. 36. The correction cannot be applied on a per bunch basis
as the LDM instrument was not operational yet for this fill. A summary of all
uncertainties is provided in Table 7. The final calibration result for
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ based on the _Track_ observable is
$\sigma_{\text{Track}}=63.0\pm 2.2\rm\,mb$. The difference with respect to the
8$\mathrm{\,Te\kern-1.00006ptV}$ calibration result is not significant given
the uncertainties, which are mostly uncorrelated.

This value is consistent with the result obtained with the BGI method
described in a previous LHCb publication [10]. The improvement in the present
result is due to the better bunch population measurement, while the
uncertainties in the overlap integral are similar.

##  7 Van der Meer scan method

###  7.1 Experimental conditions

Table 8:  Parameters of VDM scans. The scan pairs marked with an asterisk are not used in the determination of the central value of the cross-section (as explained in the text). The step duration indicates the period of stable conditions available for the measurement.  Pair | Scans | Axes | Offset | Number | Separation | Duration
---|---|---|---|---|---|---
|  |  |  | of steps | range | step (s) | total (min)
$pp$ at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$, Apr 2012, Fill 2523
1 | 2/3 | $x/y$ |  0/0 | 31/31 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 12/12
2* | 1/4 | $x^{\prime\prime}/y^{\prime\prime}$ |  0/0 | 33/31 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 13/12
3 | 5/6 | $x/y$ |  0/0 | 17/17 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 7/7
4* | 7/8 | $y/x$ |  $\sim 2\sigma_{\mathrm{n}}$/$2\sigma_{\mathrm{n}}$ | 17/17 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 7/7
$pp$ at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$, Jul 2012, Fill 2853
1 | 1/2 | $x/y$ |  0/0 | 27/27 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 12/12
2 | 3/4 | $x/y$ |  0/0 | 27/27 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 12/12
3 | 5/6 | $x/y$ |  0/0 | 27/27 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 12/12
4 | 7/8 | $x/y$ |  0/0 | 27/27 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 12/12
5* |  19/10 | $x/y$ |  0/0 | 51/51 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 21/21
6* | 11/12 | $y/x$ |  $\sim 2\sigma_{\mathrm{n}}$/$2\sigma_{\mathrm{n}}$ | 27/27 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 12/12
$pp$ at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, Oct 2011, Fill 2234
1 | 1/2 | $x/y$ |  0/0 | 31/31 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 13/13
2 | 3/4 | $x/y$ |  0/0 | 17/17 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 8/8
3 | 5/6 | $x/y$ |  0/0 | 17/17 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 15 | 8/8
$p$Pb at $\sqrt{s_{\mbox{\tiny{\it NN}}}}=5\mathrm{\,Te\kern-1.00006ptV}$, Jan
2013, Fill 3505
1 | 1/2 | $x/y$ |  0/0 | 27/27 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 6\sigma_{\mathrm{n}}$ | 30 | 17/17
2 | 3/4 | $x/y$ |  0/0 | 27/27 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 5\sigma_{\mathrm{n}}$ | 30 | 17/17
Pb$p$ at $\sqrt{s_{\mbox{\tiny{\it NN}}}}=5\mathrm{\,Te\kern-1.00006ptV}$, Feb
2013, Fill 3542
1 | 1/2 | $x/y$ |  0/0 | 27/27 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 5\sigma_{\mathrm{n}}$ | 30 | 17/17
2 | 3/4 | $x/y$ |  0/0 | 27/27 |  $\pm 6\sigma_{\mathrm{n}}$/$\pm 5\sigma_{\mathrm{n}}$ | 30 | 17/17

VDM scans at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ ($pp$) were performed
in LHCb during dedicated LHC fills in the 2012 running period, one in April
and one in July. The VDM scan method is presented in detail for these
calibrations. The calibrations at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$
($pp$) and at $\sqrt{s_{\mbox{\tiny{\it NN}}}}=5\mathrm{\,Te\kern-1.00006ptV}$
($p$Pb and Pb$p$) are summarized in Sec. 7.10. The list of dedicated
luminosity calibration fills can be found in Table 1 and the scan parameters
are listed in Table 8. Four $x$-$y$ scan pairs were performed in April and six
in July. In all scans the two beams were moved symmetrically, typically
covering a $\pm 6\sigma_{\mathrm{n}}$ range of beam separations, where
$\sigma_{\mathrm{n}}$ is the nominal beam width assuming nominal values of
$\beta^{*}$ and transverse normalized emittance
$\epsilon_{\mathrm{n}}=3.75{\,\micro\rm m}\rm\,rad$. The last scan pair in
each fill had a nominal working point at a relatively large offset $\Delta
x_{0},\Delta y_{0}\approx+2\sigma_{\mathrm{n}}$. These offset scan pairs are
only used for cross-checks and are not considered in the cross-section
determination because of their high sensitivity to systematic effects (e.g.
beam orbit drift, factorizability, linear correlations). The first and fourth
scans in April were performed along $x^{\prime\prime}$ and $y^{\prime\prime}$
axes that are rotated with respect to the principal axes of the LHC optics.
This particular scan pair is analysed, but not used in the final result.

Beam movements recorded with LHC beam position monitors (BPMs) upstream and
downstream of LHCb are shown in Fig. 37. The BPM measurements are not used in
the analysis as their time stability is insufficiently accurate. During a scan
the beam separation values are calculated on the basis of a detailed model of
the LHC magnets and are set by the accelerator control system. Dedicated
length scale calibration scans are performed in each VDM calibration fill to
experimentally verify and calibrate the beam displacements using the precisely
known geometry of the VELO (see Sec. 7.5).

Figure 37:  Beam positions recorded with the LHC beam position monitors around
the LHCb interaction point in the (left) April and (right) July VDM sessions.
The initial values are set to zero. The top (bottom) panel shows horizontal
(vertical) positions. In all scans the beams are moved symmetrically. The beam
movements around 6:30 (22:40) in April (July) correspond to the length scale
calibration by the constant separation method.

During VDM scan sessions, a large fraction of the available trigger bandwidth
was allocated to randomly triggered bunch crossings; 20${\rm\,kHz}$ were
devoted to the crossings with collisions, 2${\rm\,kHz}$ to the crossing slots
where only one of two beams was present, and 0.5${\rm\,kHz}$ to the empty
crossing slots. In addition, starting from the July 2012 session, beam-gas
events were recorded simultaneously. Due to the small beam-gas interaction
rate of a few hundred ${\rm\,Hz}$ in total, the collected events are only used
for a cross-check of the beam positions.

The average decay time of the bunch population product ${N_{1}N_{2}}$ was 36
(70) hours in the April (July) session. The value of ${N_{1}N_{2}}$ changed by
about 1% during a scan pair. Therefore, the rates are normalized by
${N_{1}N_{2}}$ of each colliding bunch pair at every scan point by defining a
specific average number of interactions per crossing

${\mu_{\mathrm{sp}}}=\frac{{\mu_{\mathrm{\mathrm{vis}}}}}{N_{1}N_{2}}\,.$ (36)

To reduce the noise associated with the $N_{1,2}$ measurements, the data for
each beam are approximated by a smoothing spline [44].

Figure 38:  Evolution of the specific average number of interactions per
crossing (${\mu_{\mathrm{sp}}}$) at the nominal head-on beam positions during
the (left) April and (right) July VDM sessions. In each scan the nominal point
was measured three times and the average over bunches is plotted with open
markers. The value at the actual zero beam separation is predicted for each
scan pair (excluding offset and tilted scans) and shown with a horizontal
band. The difference with respect to the values at zero nominal separation is
due to the working point not being exactly at zero beam separation. The beam
positions were not re-optimized during the sessions.

In addition to the bunch population changes, the luminosity stability may be
limited by changes in the bunch profiles, e.g. by emittance growth. The
luminosity stability is checked several times during the scans when the beams
were brought back to their nominal position. The evolution of
${\mu_{\mathrm{sp}}}$, averaged over bunch pairs, is shown in Fig. 38. The
average luminosity decay time is measured to be 29 (58) hours in the April
(July) session, which is largely due to the decay in the bunch population
product. The average luminosity drop caused by emittance growth (drop of
${\mu_{\mathrm{sp}}}$) amounts to 1.2% (0.5%) during the entire calibration
session in April (July). The scan points have been taken from lower to higher
$\Delta x$, $\Delta y$ values, therefore, the luminosity drop due to emittance
growth effectively enhances the rate at negative values and reduces the rate
at positive values, so that the net effect on the integrals in Eq. (15)
cancels to first order since the curves are symmetric. Therefore, the
systematic error due to the emittance growth is considered to be negligible.

As will be shown in Sec. 7.8, the measurement using the fifth scan pair in
July is more sensitive to beam orbit drifts, because it was performed at a
slightly offset working point and took longer than usual (42 instead of 24
minutes). Since the orbit drifts cannot be precisely corrected for, this scan
pair is not used for the central value of the result but is accounted for in
the systematic uncertainty. In total, for the cross-section measurement, two
_reference_ scan pairs are used from the April session and four from the July
session.

###  7.2 Overlap integral model

The rate measurements from the orthogonal VDM scans performed provide no
information on the factorizability of the single beams. However, the cross-
section measurement is sensitive to the latter. In order to impose constraints
on the factorizability, the BGI measurements performed in the same fills on
the same bunch pairs can be used. To facilitate such approach, it is
advantageous to use an identical model of the single beam densities for both
BGI and VDM analyses. In this section, a model for the overlap integral as
function of beam separation is developed based on the double Gaussian beam
shape description from Sec. 5.3. It should be noted that the application of
the classic VDM method in the case of factorizable beam distributions (see Eq.
(15)) only requires a good empirical description of the VDM scan data.

The formalism discussed in Sec. 5.2 is only valid in the case of a horizontal
beam crossing plane. However, in the April calibration the crossing plane was
rotated along $z$, which necessitates the following extended treatment. In
this more general case, the exponent in the overlap integral for Gaussian
beams from Eq. (17) gains a dependence on the product of the transverse beam
separations $\Delta x\Delta y$ and the normalization factor is modified. It is
useful to extend the definition of the effective convolved widths of the
luminous region $\Sigma_{m}$ ($m=x,y$) from Eq. (18) (taking
$\cos\phi_{m}\approx 1$) as

$\Sigma_{m}^{2}=\sigma_{m1}^{2}+\sigma_{m2}^{2}+(\sigma_{z1}^{2}+\sigma_{z2}^{2})\tan^{2}\phi_{m}\,,$
(37)

so that $\Sigma_{y}$ also has a term that depends on the corresponding half
crossing angle in the $yz$ plane $\phi_{y}$. The general expression for the
overlap integral (in the case of pure Gaussian beams) is then given by

$\displaystyle\Omega(\Delta x,\Delta y\,|\,\Sigma_{x},\Sigma_{y},\lambda)=$
$\displaystyle~{}~{}~{}\sqrt{1-\lambda^{2}}\exp\left(\frac{\lambda\Delta
x\Delta y}{\Sigma_{x}\Sigma_{y}(1-\lambda^{2})}\right)g(\Delta
x\,|\,\sigma=\Sigma_{x}\sqrt{1-\lambda^{2}})g(\Delta
y\,|\,\sigma=\Sigma_{y}\sqrt{1-\lambda^{2}})\,,$ (38)

where $g$ is the normalized Gaussian probability density function and
$\lambda$ is the correlation coefficient, which is related to the bunch
lengths and the half crossing angles by

$\lambda(\Sigma_{x},\Sigma_{y})=\frac{C_{xy}}{\Sigma_{x}\Sigma_{y}}+\bar{\lambda},\quad
C_{xy}=(\sigma_{z1}^{2}+\sigma_{z2}^{2})\tan\phi_{x}\tan\phi_{y},\quad\bar{\lambda}=\frac{\lambda_{1}+\lambda_{2}}{2}\,.$
(39)

The variables $\lambda_{1,2}$ are the individual bunch correlation
coefficients, which take into account a possible rotation of the principal
axes of the bunches (around the direction of motion). It is seen from Eq. (39)
that $C_{xy}$ is only non-zero when both $\phi_{x}$ and $\phi_{y}$ are non-
zero, which occurs when the beam crossing plane is tilted (i.e. neither
strictly horizontal nor vertical).

The overlap integral in the double Gaussian model, which is discussed in Sec.
5.3, is given by Eq. (24) with each partial overlap integral expressed as

$\Omega_{i_{x},i_{y},j_{x},j_{y}}=\Omega(\Delta x,\Delta
y\,|\,\Sigma_{x,i_{x}j_{x}},\Sigma_{y,i_{y}j_{y}},\lambda(\Sigma_{x,i_{x}j_{x}},\Sigma_{y,i_{y}j_{y}}))\,,$
(40)

where $i_{x}$, $i_{y}$, $j_{x}$, $j_{y}$ take the values $\mathrm{n}$ and
$\mathrm{w}$. Here, it is implicitly assumed that all Gaussian components have
the same centres, same bunch lengths and same correlation coefficients (i.e.
$C_{xy}$ and $\bar{\lambda}$). The weights in Eq. (24) depend on the
factorizability parameter $f_{j}$ and two weight parameters ($w_{x,j}$ and
$w_{y,j}$) for each beam $j=1,2$.

The rate that is measured in VDM scans only provides direct information on the
beam overlap integral and not on the single beam densities. Therefore, it is
convenient to parametrize the model of the overlap integral using the
effective convolved widths rather than the underlying beam width parameters.
Only three of the four effective convolved widths per plane are linearly
independent, thus two parameters are sufficient to quantify their relative
magnitudes, with the choice

$\displaystyle R_{m}$
$\displaystyle=\frac{\Sigma_{m,\mathrm{w}\mathrm{w}}}{\Sigma_{m,\mathrm{n}\mathrm{n}}}\,,\quad
R_{m}\geq 1\,,$ (41) $\displaystyle A_{m}$
$\displaystyle=\frac{\Sigma_{m,\mathrm{n}\mathrm{w}}^{2}-\Sigma_{m,\mathrm{w}\mathrm{n}}^{2}}{\Sigma_{m,\mathrm{w}\mathrm{w}}^{2}-\Sigma_{m,\mathrm{n}\mathrm{n}}^{2}}\,,\quad
A_{m}\in[-1,1]\,,$ (42)

for $m=x,y$. In terms of beam width parameters we have

$A_{m}=-\frac{(\sigma_{m1,\mathrm{w}}^{2}-\sigma_{m1,\mathrm{n}}^{2})-(\sigma_{m2,\mathrm{w}}^{2}-\sigma_{m2,\mathrm{n}}^{2})}{(\sigma_{m1,\mathrm{w}}^{2}-\sigma_{m1,\mathrm{n}}^{2})+(\sigma_{m2,\mathrm{w}}^{2}-\sigma_{m2,\mathrm{n}}^{2})}\,,$
(43)

thus $A_{m}$ describes the asymmetry between the differences in the widths of
the wide and narrow components of the two beams. The special case $A_{m}=1$
corresponds to $\sigma_{m1,\mathrm{n}}=\sigma_{m1,\mathrm{w}}$, i.e. the
distribution of beam 1 is Gaussian in the $m$ axis. In the parametrization of
Eqs. (41) and (42), the effective convolved widths are expressed as

$\displaystyle\begin{bmatrix}\Sigma_{m,\mathrm{n}\mathrm{n}}^{2}&\Sigma_{m,\mathrm{n}\mathrm{w}}^{2}\\\
\Sigma_{m,\mathrm{w}\mathrm{n}}^{2}&\Sigma_{m,\mathrm{w}\mathrm{w}}^{2}\end{bmatrix}$
$\displaystyle=\Sigma_{m,\mathrm{n}\mathrm{n}}^{2}\begin{bmatrix}r_{m,\mathrm{n}\mathrm{n}}^{2}&r_{m,\mathrm{n}\mathrm{w}}^{2}\\\
r_{m,\mathrm{w}\mathrm{n}}^{2}&r_{m,\mathrm{w}\mathrm{w}}^{2}\end{bmatrix}$
$\displaystyle=\Sigma_{m,\mathrm{n}\mathrm{n}}^{2}\left(\begin{bmatrix}1&\frac{R_{m}^{2}+1}{2}\\\
\frac{R_{m}^{2}+1}{2}&R_{m}^{2}\end{bmatrix}+A_{m}\frac{R_{m}^{2}-1}{2}\begin{bmatrix}0&1\\\
-1&0\end{bmatrix}\right)\,,$ (44)

where the ratios $r_{m,ij}$ are defined as
$r_{m,ij}\equiv\Sigma_{m,ij}/\Sigma_{m,\mathrm{n}\mathrm{n}}$.

For the actual fitting procedure, it is advantageous to use scale (or width)
parameters that have a model-independent meaning. For example, the RMS of the
luminosity profiles at $\Delta x=0$ and $\Delta y=0$ are such parameters,
which can be easily estimated from the scan data to obtain starting values for
the fit. Using the parameters defined above, the RMS $S_{x}$ of the luminosity
profile at $\Delta y=0$ is given by

$\displaystyle S_{x}^{2}$ $\displaystyle=\frac{\int\Delta x^{2}\Omega(\Delta
x,0)d\Delta x}{\int\Omega(\Delta x,0)d\Delta
x}=\Sigma_{x,\mathrm{n}\mathrm{n}}^{2}\frac{\sum_{I}\frac{w_{I}}{r_{y,i_{y}j_{y}}}r_{x,i_{x}j_{x}}^{2}(1-\lambda^{2}(\Sigma_{x,\mathrm{n}\mathrm{n}}r_{x,i_{x}j_{x}},\Sigma_{y,\mathrm{n}\mathrm{n}}r_{y,i_{y}j_{y}}))}{\sum_{I}\frac{w_{I}}{r_{y,i_{y}j_{y}}}}$
(45)

and similarly for the other coordinate. The values
$\Sigma_{x,\mathrm{n}\mathrm{n}}$ and $\Sigma_{y,\mathrm{n}\mathrm{n}}$ are
obtained by solving the system of equations defined by the above equation.
Finally, in the double Gaussian model, the shape of the overlap integral as
function of beam separation is parametrized with the following 14 parameters

$S_{x},S_{y},R_{x},R_{y},A_{x},A_{y},f_{1},f_{2},w_{x,1},w_{y,1},w_{x,2},w_{y,2},C_{xy},\bar{\lambda}\,.$
(46)

Only some of those parameters remain free for the cross-section determination
as explained in the following section.

###  7.3 Cross-section determination

All colliding bunch pairs are analysed individually. The data are fitted
simultaneously for pairs of $x$ and $y$ scans. The value of
$\mu_{\mathrm{vis}}$ at each step $k$ is described with

$\mu_{\mathrm{vis}}(\Delta x_{k},\Delta
y_{k})=\sigma_{\mathrm{vis}}\,N_{1}N_{2}\,\Omega(\Delta x_{k}-\Delta
x_{0},\Delta y_{k}-\Delta
y_{0})+N_{1}\mu_{\mathsf{be}}^{\mathrm{sp}}(t_{k})+N_{2}\mu_{\mathsf{eb}}^{\mathrm{sp}}(t_{k})\,,$
(47)

where $N_{1,2}$ are the bunch intensities and
$\mu_{{\mathsf{be}}({\mathsf{eb}})}^{\mathrm{sp}}$ is the specific $\mu$ value
of the beam-gas background for beam 1(2). Two position parameters, $\Delta
x_{0}$ and $\Delta y_{0}$, were introduced to account for the fact that the
luminosity may reach a maximum at a non-zero nominal separation ($\Delta x$,
$\Delta y$) due to an imperfect alignment of the beams.

The last two terms in Eq. (47) are due to beam-gas interactions and are
proportional to the beam intensity and the residual gas pressure. The value of
$\mu_{{\mathsf{be}}({\mathsf{eb}})}^{\mathrm{sp}}$ is determined using events
from data taken simultaneously for non-colliding bunches for each scan
independently and is typically $2\times 10^{-14}$ ($1\times 10^{-14}$). In
most cases the pressure was very stable during VDM scans. During the first
pair of scans in the July session a drop of about 10% was observed due to the
neon gas injection system being used beforehand. In order to take this into
account, an exponential dependence on time is assumed for the first pair of
scans in July, while in all other cases constant specific background is
assumed.

As already discussed above, due to the pattern of two orthogonal movements
used in the scans, the data provide no information on the factorizability
parameters and the linear correlation parameters. Therefore, in order to
obtain the required additional information, we use the BGI measurements
performed in the same fills as the VDM scans. The BGI analysis gives $f_{1,2}$
values compatible with zero. In the VDM analysis the $f_{1,2}$ parameters are
fixed to zero. In this fully non-factorizable case, the two weight parameters
$w_{x}$ and $w_{y}$ for each beam are perfectly anticorrelated as seen from
Eq. (23). Therefore, only one weight parameter per beam is used for the VDM
analysis. The value of $C_{xy}$ is computed using the BGI measurements of the
convolved bunch length and crossing angles. No significant linear correlation
in the beam distributions is observed, thus the value of $\bar{\lambda}$ is
set to zero. The available number of counts per scan point and the relatively
coarse scanning grid do not allow measuring both $R_{m}$ and $A_{m}$. The
parameter $R_{m}$ describes the main features of $\Sigma_{m,ij}$, Eq. (7.2),
and is determined in the fit, while $A_{m}$ is fixed to zero for the
determination of the central value of $\sigma_{\mathrm{vis}}$. The systematic
uncertainties arising from the assumptions on the fit parameters are discussed
in Sec. 7.7. In total nine parameters remain free, including the visible
cross-section.

Initially, the fit is performed using the uncertainty estimates obtained from
the data ${\delta\mu_{\mathrm{vis}}=\sqrt{1/N_{0}-1/N}}$, where $N_{0}$ is the
number of empty events in a total of $N$ events. For small values of
$\mu_{\mathrm{vis}}$, these uncertainty estimates are correlated with the
$\mu_{\mathrm{vis}}$ values themselves, thus biasing the data weights and the
fit result. To mitigate this problem, the data are fitted a second time using
uncertainties
$\delta\mu_{\mathrm{vis}}=\sqrt{\exp(\hat{\mu}_{\mathrm{vis}})/N-1/N}$, which
are based on the predictions $\hat{\mu}_{\mathrm{vis}}$ of the first fit.

Figure 39:  Fit of VDM profiles for a single bunch and scan pair in the (top)
April and (bottom) July VDM sessions. The left (right) panels show data and
fit predictions corresponding to the $x$ ($y$) scan. The data points are
represented with circles, while the fitted curve is shown as a solid line. The
error bars are smaller than the symbol size. The fit pulls are displayed below
each fit projection and show no systematic structure. There are three data
points at $\Delta m=0$ as the nominal point was measured three times for each
scan.

Figure 40:  (Top) measured cross-section values and (bottom) the corresponding
values of $\chi^{2}/\mathrm{ndf}$ for all bunch pairs and for non-offset scan
pairs in the (left) April and (right) July VDM sessions. Measurements within
one scan pair are separated from the rest by a larger distance.

For presentation purposes we define a corrected specific interaction rate per
crossing by rearranging Eq. (47)

$\mu_{\mathrm{vis},i}^{\mathrm{sp}}=\frac{\mu_{\mathrm{vis}}(\Delta
x_{i},\Delta
y_{i})-N_{1}\mu_{\mathsf{be}}^{\mathrm{sp}}(t_{i})-N_{2}\mu_{\mathsf{eb}}^{\mathrm{sp}}(t_{i})}{N_{1}N_{2}}\,.$
(48)

Then, the function $\sigma_{\mathrm{vis}}\Omega(\Delta x,\Delta y)$ represents
the fit to the corrected data points $\mu_{\mathrm{vis},i}^{\mathrm{sp}}$. An
example of a fit for a single bunch pair is shown in Fig. 39 for one scan
pair. The measured cross-section values and the values of $\chi^{2}$ per
degree of freedom, $\chi^{2}/\mathrm{ndf}$, obtained for all bunch and scan
pairs are shown in Fig. 40. Higher values of $\sigma_{\mathrm{vis}}$ are
obtained for the fifth scan pair in July. The associated uncertainty is
discussed in detail in Sec. 7.8. For the April scan session, the values of
$\chi^{2}/\mathrm{ndf}$ are on average higher than one. This can be explained
by the smaller beam size and the fact that the uncertainty of the beam
separation is not taken into account. Finally, the cross-section is obtained
for each calibration session by calculating the weighted average of all
measurements from reference scan pairs.

The sources of systematic effects that influence the result of the fit
described with Eq. (47) can be grouped into two major categories. First, there
can be systematic effects related to the inputs of the fit, namely bunch
intensities (described in Sec. 3), rates (of signal and background) and beam
positions (Secs. 7.4–7.6). Second, the influence of the VDM profile model, the
technical aspects of the fitting procedure and discrepancies between repeated
measurements are described in Secs. 7.7 and 7.8. The variation in the obtained
cross-section among bunch pairs is used to estimate the uncertainty due to the
relative bunch intensity measurement. The uncertainty due to a potential non-
linearity of the FBCT device is estimated to be about $0.05\%$ using the same
method as described in Sec. 6.6. Moreover, by using the independent
measurements from the ATLAS BPTX system [38], a discrepancy in the final
result of about $0.1\%$ is found, which is assigned as an additional
systematic uncertainty.

###  7.4 Rate measurement

For the absolute calibration, the beam-gas related backgrounds are subtracted
taking into account the difference in bunch populations. There is a
statistical uncertainty associated with the measured specific background per
proton used for subtraction. The value of the specific background is shared
among cross-section fits of bunch pairs and introduces a correlation, which is
taken into account when combining the measurements. This uncertainty is
estimated to be less than 0.1%.

The beam-beam related background is estimated to be 0.1–0.2% by taking the
difference between the visible cross-sections of the restricted and the non-
restricted _Track_ observables. For the central value of the visible cross-
section we use the less affected _Track_ observable, while the full difference
is taken as systematic uncertainty.

In the presence of a non-zero beam crossing angle, the luminous region
position varies with transverse beam separation. The track reconstruction
efficiency is not uniform as function of the primary vertex longitudinal
position. Therefore, a correction to the $\mu_{\mathrm{vis}}$ values is
applied using the same principle as described in Sec. 4.

Figure 41:  Luminous region longitudinal position and size as function of beam
separation for a scan pair in the (left) April and (right) July VDM sessions.
The values for other scan pairs are similar. Markers indicate the mean while
the bands between dotted lines contain 68% of the vertices. In April, the
luminous region moves in both horizontal and vertical scans because of the
crossing angle configuration. The inversion of the slope in the horizontal
scan is due to the change of sign of the crossing angle between the two
sessions.

Figure 42:  Efficiency correction factors for the _Track_ observable for a
scan pair in the (left) April and (right) July VDM sessions. The values for
other scan pairs are similar.

The longitudinal position and size of the luminous region are measured for
each step using a single Gaussian fit to the selected beam-beam vertices,
taking into account the vertex reconstruction efficiency as a function of $z$.
The quantities are linearly extrapolated to large separations, where the
luminosity vanishes and not enough vertices are available ($<1000$). The
measured longitudinal parameters of the luminous region are shown in Fig. 41
and the calculated correction factors to the observed $\mu_{\mathrm{vis}}$ for
each scan and step are shown in Fig. 42. It can be noted that the correction
factors are approximately linear as function of separation. Therefore, no
large effect on the VDM profile width is expected. The effect of the
correction amounts to $+0.32\%$ ($-0.03\%$) in the April (July) calibration.
An estimate of the associated systematic uncertainty is made using a
comparison with an unaffected observable (_Calo_). In the approximation of
pure Gaussian beams, the rate at zero beam separation is proportional to
$\sigma_{\mathrm{vis}}/(\Sigma_{x}\Sigma_{y})$, see Eqs. (1), (2) and (17).
Equivalently, the measured cross-section is proportional to the rate at zero
separation and the product of the widths of the VDM profiles. It is useful to
separate the correction factor dependence on $\xi_{lz}$ (Fig. 10) in a linear
and a higher order part. The former only affects the rate at zero separation,
while the latter mainly affects the widths. The residual (after correction)
slope of the ratio ${\mu_{\mathrm{Track}}}/{\mu_{\mathrm{Calo}}}$ as function
of $\xi_{lz}$ is measured and normalized to the ratio at $\xi_{lz}=0$. The
$\xi_{lz}$ value at the working point is multiplied by that slope to obtain
the uncertainty on the cross-section due to the linear part of the correction.
The uncertainty due to the higher order part of the correction is estimated by
comparing the product $\Sigma_{x}\Sigma_{y}$ measured using the _Track_ and
the _Calo_ observable. The two uncertainties are summed linearly to obtain the
systematic uncertainty of 0.13% (0.07%) for the April (July) calibration.

###  7.5 Length scale

The nominal beam separation values $\Delta x$ and $\Delta y$ are calculated
from the LHC magnet currents at every scan step. An absolute calibration of
the beam separation is made using the more precise VELO scale. In principle,
each beam can have different calibration constants. However, since for all
scans the beams were moved symmetrically (as opposed to one beam at a time),
only the average (or common) length scale matters to first order for
calibrating the separation. During the length scale calibration scans the
beams were moved in several steps, with each measurement lasting from one to
four minutes. The time intervals during which the beams were stationary are
determined using the recorded states of the deflection magnets in the LHC
logging data. Two methods are used in order to obtain the absolute length
scale calibration.

##### Constant beam separation method:

Both beams are moved in five equidistant steps in $\Delta x$ and $\Delta y$
keeping their nominal separation constant. During a simultaneous parallel
translation of both beams, the centre of the luminous region should follow the
beam positions regardless of the bunch shapes. The luminous region centre can
be determined using beam-beam vertices measured with the VELO.

Beam-beam vertices are selected using randomly triggered events in
${\mathsf{bb}}$ crossings. In the 2013 calibrations the average number of
interactions per crossing was very low. Therefore, the standard trigger
requirement for beam-beam events in ${\mathsf{bb}}$ crossings was used in
order to collect a sufficient amount of data. To remove background from
material interactions, only vertices that lie within $2\rm\,mm$ from the
median in each axis are retained. Potential beam-gas interaction background is
reduced by applying a loose cut on the longitudinal position of the vertices
$|z|<350\rm\,mm$ and by requiring at least one track in each direction. The
vertex coordinates are binned in $x$ and $y$. The bin width is chosen
automatically based on the data according to the Freedman–Diaconis rule.777
The bin width is determined as two times the interquartile range (the
difference between the 75th and the 25th percentiles) divided by the number of
observations in the sample to the power of $1/3$ [45].  Data that deviates
more than $5\sigma$ from the median is discarded.

An empirical model is fitted to the histogram for each coordinate and the mean
of the model distribution is used as an estimator for the position of the
luminous region. A sum of two Gaussian functions, where the two means are not
required to be equal, is found to fit the data well in all cases. The
distribution of the luminous region is not expected to change during the scans
if the beam separation is kept constant. Using this property as a constraint,
an additional global fit per scan is performed, which has a single set of
shape parameters with the exception of the mean, which is independent for each
step.

A simultaneous drift of both beams in the same direction affects the length
scale measurement. The drift is estimated using the steps before and after the
scans, when the beams are nominally centred. Differences of the luminous
region position before and after a scan are attributed entirely to
simultaneous beam drift. A correction is applied for each step during the scan
by using a linear interpolation. Since the beam separation is kept constant,
beam-beam effects do not influence the measurement.

The measured luminous region position for each step is fitted against the
average of the nominal beam positions using a weighted linear fit. The fit is
performed simultaneously for all colliding bunch pairs using independent
intercepts and a common slope. The results of this fit are shown in Fig. 43.
The slope parameter of the fit is an estimate of the length scale calibration
factor.

Figure 43:  Length scale calibration fits in the (left) $x$ and (right) $y$
axis in the July VDM session. The fitted intercepts are subtracted from both
the fit curves and the data points as they have no significance. Each data
point is an aggregate of the measurements from 16 colliding bunch pairs for
the constant separation method and 35 individual bunches for the BGI method.
The fit slope obtained using the constant separation method (solid line)
should be compared with the average of the slopes obtained with the BGI method
(dashed and dotted lines).

##### Beam-gas imaging method:

Individual beam positions are measured using beam-gas interactions. The neon
gas injection system was used to enhance the beam-gas interaction rate. The
beams were moved similarly to VDM scans, but in a few large steps. Beam-gas
vertices for each beam are selected using the corresponding standard trigger
requirements for beam-empty and beam-beam crossings. In order to remove
background from material interactions, only vertices that lie within
$2\rm\,mm$ from the median in each axis are retained. Potential beam-beam
interaction background is reduced by requiring no tracks in the opposite
direction of the corresponding beam. Consistent comparison between
measurements from colliding and non-colliding bunches is ensured by imposing a
cut on the longitudinal position of the vertices $|z|>300\rm\,mm$. To obtain
the single-beam offsets, the $x$ and $y$ coordinates of the beam-gas vertices
are fitted with a straight line as function of $z$. The weight assigned to the
individual vertices takes into account the beam width and the vertex
resolution.

Individual beam positions are measured before and after the scans, when the
beams are nominally centred. No significant beam drifts are observed at the
$0.5{\,\micro\rm m}$ level, thus no correction is made. The beam-beam
deflection effect influences beam positions in ${\mathsf{bb}}$ bunch
crossings. We take this into account as described in Sec. 7.6, where $\Sigma$
values are taken from the VDM scan measurements. Taking this effect into
account improves the consistency between measurements from colliding and non-
colliding bunches.

The measured beam position for each step is fitted against the nominal setting
using a weighted linear fit. The fit is performed simultaneously for all
bunches in each beam using one slope and one intercept parameter. The results
of this fit are shown in Fig. 43. To obtain an effective common length scale,
the average between both beams is taken.

Table 9:  Length scale calibration constants. All uncertainties are statistical.  Calibration | Method | $\left(\frac{\Delta x(\mathrm{VELO})}{\Delta x(\mathrm{LHC})}-1\right)\times 10^{2}$ | $\left(\frac{\Delta y(\mathrm{VELO})}{\Delta y(\mathrm{LHC})}-1\right)\times 10^{2}$
---|---|---|---
April 2012 | Constant separation | $-1.10\pm 0.02$ | $-0.30\pm 0.02$
July 2012 | Constant separation | $-0.14\pm 0.03$ | $\phantom{-}0.54\pm 0.04$
July 2012 | Beam-gas imaging | $\phantom{-}0.29\pm 0.06$ | $\phantom{-}0.69\pm 0.05$

The length scale calibration constants are summarized in Table 9. A
statistically significant discrepancy of about $0.5\%$ is observed between the
calibration factors for the $x$ axis obtained using the two methods in July.
The origin of this difference is not understood. Therefore, we assign $0.5\%$
uncertainty to all length scale calibrations. It is assumed that this
uncertainty is correlated between calibrations. The statistical uncertainties
are small compared to the discrepancy and are neglected. The calibration
constants used for July are those obtained by the constant separation method.
Thus, we maintain consistency of treatment with other calibration sessions,
for which we have not performed a beam-gas imaging calibration.

###  7.6 Beam-beam effects

The electromagnetic interaction between charged particles of two colliding
bunches is called _beam-beam effect_. There are two aspects of this
interaction that affect the VDM scan measurements. The first, called _dynamic
$\beta$ effect_, is the result of the mutual (de)focusing of the two colliding
bunches. It leads to a change in $\beta^{*}$ (and thus beam size), which
depends on the transverse beam separation. Therefore, the transverse size of
the bunches is not constant during the VDM scans. Secondly, the closed orbits
of the bunches are distorted by the angular kick induced by their
electromagnetic repulsion. This _beam-beam deflection_ effect has a different
magnitude depending on transverse separation, thus distorting the scan
profiles.

The so-called _beam-beam parameter_ that quantifies the strength of the
dynamic $\beta$ effect in the axis $m$, for a normally distributed bunch in
beam $j$ is given by [46]

$\xi_{m,j}=\frac{\alpha\hbar
c}{2\pi}\frac{N_{j^{\prime}}Z_{j^{\prime}}\beta^{*}_{m,j}}{E_{j}\sigma_{m,j^{\prime}}(\sigma_{x,j^{\prime}}+\sigma_{y,j^{\prime}})}\xi_{m}^{\mathrm{rel}}(\textstyle{\frac{\Delta
x}{\sigma_{x,j}}},\textstyle{\frac{\Delta y}{\sigma_{y,j}}})\,,$ (49)

where $j^{\prime}$ denotes the bunch in the opposite beam,
$N_{j^{\prime}}Z_{j^{\prime}}$ is the bunch charge in units of elementary
charge, $\beta^{*}_{m,j}$ is the value of the $\beta$ function at the IP,
$E_{j}$ is the ring energy setting (particle energy divided by particle
charge) and $\sigma_{m,j^{\prime}}$ are the bunch sizes. The function
$\xi_{m}^{\mathrm{rel}}$ was modelled [47] using the MAD-X optics software
[48] and is shown in Fig. 44. It is independent of the machine optics
parameters and the bunch properties. For small separations $\Delta x$ and
$\Delta y$ the value of the $\xi_{m}^{\mathrm{rel}}$ is close to unity and
approximately constant. Therefore, the beam-beam force is approximately
linear, resembling the force of a quadrupole field. However, the force becomes
non-linear for large separations. The above equation is also valid in the case
of ion beams.

Figure 44:  Relative change of the beam-beam parameter as function of beam
separation. The values are obtained using a simulation of the machine optic
elements.

The ratio between the perturbed $\beta^{*}$ and the unperturbed
$\beta^{*}_{0}$ as function of separation is given by

$\frac{\beta^{*}_{m,j}}{\beta^{*}_{0,m,j}}=(1+4\pi\xi_{m,j}\cot(2\pi
Q_{m,j})-4\pi^{2}\xi_{m,j}^{2})^{-1/2}\,,$ (50)

where $Q_{m,j}$ is the machine tune of beam $j$ in the axis $m$. While there
is a recursive dependence between $\xi$ and $\beta^{*}$, this is only a second
order effect and can be neglected by using the nominal $\beta^{*}_{0}$ in Eq.
(49). Possible collisions at other IPs also contribute to the dynamic $\beta$
independently from the local beam separation. Such contributions, provided
beams are not moving at other IPs, are constant and effectively modify the
nominal value of $\beta^{*}$. Since these modifications are much smaller than
the uncertainty in the value of the nominal $\beta^{*}$ itself, they are
neglected.

To calculate the effect on the luminosity, the relative change of $\beta^{*}$
is computed with respect to a reference value. Applying a correction to the
rate (or the overlap integral) effectively removes the dependence of bunch
shapes on separation. The choice of the reference $\beta^{*}$ value is
arbitrary and determines the values of the effective separation-independent
bunch parameters. It can be shown that this choice has a negligible impact on
the net correction. It is beneficial to use $\beta^{*}(0,0)$ as the reference,
in order to enable consistent comparisons of VDM profile parameters from
different scans. The relative $\beta^{*}$ change (see Fig. 45) is calculated
for each colliding bunch pair using the parameters of the nominal optics, the
measured bunch intensities and the convolved widths. In this case, the shape
of the bunches is approximated with a single Gaussian distribution and the
size of the colliding bunches in each pair is assumed equal. The relative
change of the luminosity is obtained using the perturbed bunch widths.
Finally, the correction factor to the rate is obtained, which is shown in Fig.
46.

Figure 45:  Relative $\beta^{*}$ change for beam 1 in the $x$ axis as as
function of beam separation for a scan pair in the (left) April and (right)
July VDM sessions. The values for other scan pairs are similar. The values for
beam 2 are similar owing to similar bunch intensities. The values for the $y$
axis are also similar with $x$ and $y$ scans exchanged. The markers and the
solid line represent the average, while the dotted lines correspond to the
extremes among colliding bunches. The reference $\beta^{*}$ is chosen such
that the relative change is zero at the separation corresponding to maximum
luminosity.

Figure 46:  Rate correction factor for the dynamic beta effect for a scan pair
in the (left) April and (right) July VDM sessions. The values for other scan
pairs are similar. The markers and the solid line represent the average, while
the dotted lines correspond to the extremes among colliding bunches. Owing to
the dynamic $\beta$ effect, the $\beta$’s and thus the beam sizes depend on
the beam separation. To take this into account, the effect on the luminosity
is compensated by multiplying the observed rates with the correction factors.

The beam-beam angular kick causes a deflection of beam $j$, which is
calculated numerically using the formalism of Ref. [49]

$\theta_{y,j}+i\,\theta_{x,j}=-2\alpha\hbar
c\frac{N_{j^{\prime}}Z_{j^{\prime}}}{E_{j}}F_{0}(\Delta x,\Delta
y,\Sigma_{x},\Sigma_{y})\,,$ (51)

where $N_{j^{\prime}}Z_{j^{\prime}}$ is the bunch charge in the opposite beam
in units of elementary charge, $i$ is the imaginary unit, $F_{0}$ is a complex
function and Gaussian bunch profiles are assumed. The shift of the closed
orbit in the $m$ coordinate for beam $j$ is given by

$\delta_{m,j}^{\mathrm{BB}}=\theta_{m,j}\beta^{*}_{m,j}\frac{1}{2\tan(\pi
Q_{m,j})}\,,$ (52)

while the effect on the beam separation is the sum of the individual beam
shifts

$\delta^{\mathrm{BB}}\Delta
m=\delta_{m,1}^{\mathrm{BB}}+\delta_{m,2}^{\mathrm{BB}}\,.$ (53)

The corrections that are added to the nominal beam separation are shown in
Fig. 47. The procedure outlined above was verified by the direct observation
of beam-beam deflections with LHC orbit data [50], which were found to agree
well with the expectations.

Figure 47:  Correction to the nominal separation in the $x$ axis for beam-
beam-induced orbit shift for a scan pair in the (left) April and (right) July
VDM sessions. The values for other scan pairs are similar. The values for the
$y$ axis are similar with $x$ and $y$ scans exchanged. The markers and the
solid line represent the average, while the dotted lines correspond to the
extremes among colliding bunches.

The systematic uncertainty of the correction for beam-beam effects is
estimated by numerically propagating the uncertainties of the input
parameters. In addition to $\beta^{*}$ and machine tune, uncertainties are
assigned to the ratio of bunch widths and the convolved bunch width to account
for the assumption of identical bunch shapes and Gaussian bunches,
respectively. The assumed uncertainties on input parameters and their
correlations are listed in Table 10. The uncertainty is similar for the two
8$\mathrm{\,Te\kern-1.00006ptV}$ $pp$ calibrations and amounts to about
$0.3\%$.

Table 10:  Uncertainties of input parameters to the correction for beam-beam effects. All quantities are correlated between axes. The quantities $\beta^{*}_{m,j}$ and $Q_{m,j}$ are not correlated between the two beams.  Source | $\beta^{*}_{m,j}$ | $Q_{m,j}$ | $\sigma_{m,1}/\sigma_{m,2}$ | $\Sigma_{m}$
---|---|---|---|---
Uncertainty | 20% | 0.02 | 50% | $\sim 10\%$

###  7.7 Fit model uncertainty

The choice of a particular model to describe the single beam shapes gives rise
to a systematic uncertainty. The bias in the case of the BGI method is studied
by simulating various deviations of the single-beam distributions from the
assumed model as described in Sec. 6.6. We assume that the same uncertainty of
$0.5\%$ applies to the VDM calibration and that it is fully correlated between
the methods. In addition, the fitting procedure applied to VDM scan data might
introduce a bias. The potential fit bias is estimated to be less than $0.2\%$
by applying the analysis to simulated VDM scans with experimental conditions
and beam properties similar to those of data.

The linear correlations in VDM scan profiles can be non-zero due to a tilt of
the beam crossing plane or due to non-zero linear correlations with respect to
the scanning axes of the transverse bunch distributions in Eq. (39). The
latter are found to be small and are neglected. The bunch crossing plane is
nominally horizontal in all calibration fills, except in April, when the tilt
was approximately $-21.5\degree$. Taking into account the correlation, the
April calibration result changes by about $+0.3\%$. The effect of the
correlation is most pronounced if the scan pair crossing point is not centred
at zero, as seen from the exponent factor in Eq. (7.2). For the nominal scans,
the correction from the exponent factor is negligible. However, the correction
to the fourth (offset) scan pair in April amounts to $-8\%$, improving
significantly the consistency with the reference scans. Rather than
propagating the uncertainty of the fixed parameter $C_{xy}$ to the fit result,
half of the effect of the correction is used as an estimate of the systematic
uncertainty.

The major uncertainty of the VDM calibration is due to the assumptions made on
the parameter values as described in Sec. 7.3. In order to estimate the
corresponding bias, the fits are redone without fixing parameters but using
constraints obtained from BGI measurements. The information that is used from
the BGI is limited to the factorizability of the beams, thus the VDM
calibration is as independent as possible. In particular, constraints are not
applied directly using dimensional parameters (widths) from BGI measurements,
which are subject to resolution and alignment uncertainties. Moreover, for
each scan pair, all bunch pair fits are subject to the same constraints, with
their width taking into account the spread of the parameter values among bunch
pairs.

First of all, Gaussian constraints on the factorizability parameters $f_{1,2}$
are added to the fit. However, these constraints are not sufficient to ensure
that the beams are non-factorizable. This is easily seen by considering sets
of degenerate parameters for which the beam distribution is Gaussian in at
least one of the axes. Therefore, a generic measure of the transverse
factorizability is introduced as

$v_{j}=1-\frac{\int\rho_{j}(x,y=0)dx\int\rho_{j}(x=0,y)dy}{\rho_{j}(x=0,y=0)}\,,$
(54)

where $\rho_{j}$ is the transverse beam distribution of beam $j=1,2$. The
measure $v_{j}$ is zero if $\rho_{j}$ is factorizable. The values of $f_{1,2}$
and $v_{1,2}$ measured with BGI, as well as the predicted values at the time
of the VDM scans are shown in Fig. 48.

Figure 48:  Values of the (top) factorizability parameter $f_{j}$ and (bottom)
generic factorizability measure $v_{j}$ in the (left) April and (right) July
VDM sessions. The filled markers and the corresponding error bars indicate the
average and the RMS of BGI measurements for individual bunch pairs. In the
case of the July scans, BGI measurements are available before and after the
VDM session, thus the values are linearly interpolated to obtain predictions
at the time of VDM scan pairs (open markers). For April, as BGI data are only
available after the VDM scans, the closest data points are used as a
prediction, while the difference to the linear extrapolation is added in
quadrature to the uncertainties. The trend of the generic factorizability
measure shows that beam distributions become more factorizable as function of
time.

The values of $v_{1,2}$, being functions of the individual beam distributions,
cannot be computed using only the parameters describing the overlap integral
from Eq. (46). Therefore, two additional parameters for each coordinate
$m=x,y$ are added to the overlap fit model, which are sufficient to compute
$v_{1,2}$:

$\displaystyle C_{m}$
$\displaystyle=(\sigma_{z1}^{2}+\sigma_{z2}^{2})\tan^{2}{\phi_{m}}$ (55)
$\displaystyle a_{m}$
$\displaystyle=\frac{\sigma_{m1,\mathrm{n}}^{2}}{\sigma_{m1,\mathrm{n}}^{2}+\sigma_{m2,\mathrm{n}}^{2}}=\frac{\sigma_{m1,\mathrm{n}}^{2}}{\Sigma_{m,\mathrm{n}\mathrm{n}}^{2}-C_{m}}\,,\quad
a_{m}\in[0,1]\,.$ (56)

Gaussian constraints of 10% around the measured value from BGI are applied to
$C_{x,y}$. Additionally, to avoid unphysical sets of beam parameters, beams
are required to have similar transverse sizes in each coordinate. This
requirement is ensured by imposing a weak Gaussian constraint of $0\pm 0.1$ on
the asymmetry of the RMS of the two beam distributions. Such constraints are
justified by the equal design $\beta^{*}$ and emittance of the two beams. The
asymmetries estimated from the BGI measurements for all bunch pairs lie within
$\pm 1\sigma$, indicating that the width of the constraints is sufficiently
large.

The VDM data for each bunch and scan pair are fitted using the additional
parameters and the constraints described above. Moreover, profiles of
$\chi^{2}/\mathrm{ndf}$ are obtained as function of $\sigma_{\mathrm{vis}}$.
The minima of the $\chi^{2}/\mathrm{ndf}$ profiles are very close to the
values of $\sigma_{\mathrm{vis}}$ obtained with the fit where
$\sigma_{\mathrm{vis}}$ is a free parameter as seen in Fig. 49.

Figure 49:  Examples of the constrained fit $\chi^{2}/\mathrm{ndf}$ profiles
as function of $\sigma_{\mathrm{vis}}$ for (left) scan pair 1 in April and
(right) scan pair 2 in July. Each curve corresponds to one bunch pair. The
values of $\sigma_{\mathrm{vis}}$ obtained with direct fitting are shown with
cross markers.

The new values of $\sigma_{\mathrm{vis}}$ are averaged for each scan pair and
the obtained value is compared to the baseline cross-section for that pair. A
difference of $0.6\%$ and $0.9\%$ is found for the April and the July
calibration, respectively. The difference is attributed to the fact that
$f_{1,2}$ and $A_{x,y}$ are fixed for the baseline fits and is assigned as a
systematic uncertainty.

Another systematic uncertainty arises from the uncertainty in the central
values of the constraints obtained from BGI analysis. To estimate this
uncertainty, the constraints on $f_{1,2}$ and $v_{1,2}$ are varied by one
standard deviation. The cross-section obtained with the modified constraints
is compared to that obtained with the nominal constraints. The difference of
$0.3\%$ (for both April and July calibrations) is taken as an additional
systematic uncertainty, which is considered fully correlated with the BGI
result. The average $\chi^{2}/\mathrm{ndf}$ profiles for the reference scan
pairs are shown in Fig. 50 together with the relative contributions of the VDM
data and the constraints.

Figure 50:  Average $\chi^{2}/\mathrm{ndf}$ profiles as function of the
relative difference to the baseline cross-section for the reference scan pairs
in (top) April and (bottom) July shown with the top pair of lines (blue). The
profiles corresponding to the nominal and modified constraints are shown with
solid and dashed lines, respectively. The minima of the
$\chi^{2}/\mathrm{ndf}$ profiles are marked with a cross. The contribution of
the VDM data, $\chi^{2}_{\mbox{\tiny VDM}}$, normalized to the number of
points in the VDM scan pair $n_{\mbox{\tiny VDM}}$ is shown with the middle
pair of lines (red). The contribution of the constraints,
$\chi_{\mathrm{c}}^{2}$, normalized to the number of constraints
$n_{\mathrm{c}}$ is shown with the bottom pair of lines (yellow).

###  7.8 Reproducibility

An important source of non-reproducibility of VDM scan measurements is the
drift of the beam orbits. In addition, in the presence of a non-zero crossing
angle, a drift in the longitudinal bunch crossing point, $z_{\rm rf}$, can
influence the beam separation. While the effect of the latter is found to be
negligible, the former has potentially a sizeable effect, which cannot be
reliably corrected. For the following discussion data are averaged over bunch
pairs since these effects are common for all bunches.

The relation between the position of the luminous region and the beam
separation from Eq. (19) can be used to estimate orbit drifts and $z_{\rm rf}$
values in the case of non-zero crossing angle. In the general case of a
crossing angle in both $xz$ and $yz$ planes, the luminous region centre for
Gaussian beams is

$\xi_{lz}=z_{\rm rf}+k\Delta x^{\prime}\,,$ (57)

where $k$ contains the beam widths and angles and $\Delta x^{\prime}$ is the
separation in the crossing plane, which is given by

$\Delta x^{\prime}=\Delta x\cos\psi+\Delta
y\sin\psi\,,\quad\tan\psi=\frac{\tan\phi_{y}}{\tan\phi_{x}}\,.$ (58)

Here $\psi$ is the tilt of the crossing plane, which is approximately
$-21.5\degree$ in April 2012 and negligible for all other calibrations.

Figure 51:  Values of $z_{\rm rf}$ in (left) April and (right) July scans. The
implied assumptions are that $z_{\rm rf}$ does not change and there is no
orbit drift during a scan pair. No measurements are performed for offset
scans. The deviations of the measurements from the weighted average
(horizontal dotted line) are not statistically significant.

In accordance with Eq. (19), the value of $z_{\rm rf}$ is measured by
interpolating the longitudinal position of the luminous region $\xi_{lz}$ at
the separation that corresponds to the maximum luminosity. It is implicitly
assumed that $z_{\rm rf}$ does not change and that there is no orbit drift
during a scan pair. When the scanning axis is orthogonal to the beam crossing
plane (e.g. $y$ scans in July) this measurement cannot be made, as $\xi_{lz}$
is not expected to change during the corresponding scan. The measured $z_{\rm
rf}$ values, which are shown in Fig. 51, are found to be consistent between
scans and with those obtained with the BGI analysis in the same fills.
Therefore, it is assumed that $z_{\rm rf}$ values do not vary during a scan
and the associated uncertainty is negligible.

The overall drift of the beam separation between scan pairs is automatically
taken into account in the VDM analysis, since it is effectively only a shift
of the nominal separation where the luminosity is maximal. However, the drift
during a scan pair can introduce a bias to the measurement. It is useful to
split the drift into two components: a “slow” one that corresponds to the time
scale of a scan pair and a “fast” one that corresponds to the time scale of a
step.

Figure 52:  Estimated “slow” drift of the beam separation in the (left) April
and (right) July VDM sessions. The nominal separation where the luminosity is
maximal gives a single estimate per scan of the drift in the corresponding
direction (circles). The uncertainties are smaller than the marker size. The
data for each coordinate are fitted with a smooth function (curve) and a
prediction is made at each step of every scan (solid curve).

The slow beam separation drift can be estimated from the fitted position of
the VDM profile maximum. One estimate per coordinate is obtained for each scan
pair, as shown in Fig. 52.

Figure 53:  Longitudinal position of the luminous region as function of
separation in the crossing plane for reference scans in the (left) April and
(right) July VDM sessions. It is assumed that there is no significant orbit
and $z_{\rm rf}$ drift during the reference scans. The data for each scan
direction are fitted with a smooth function. In July the crossing plane is
orthogonal to the $y$ axis, thus $\xi_{lz}$ is not expected to change during
the $y$ scans and the data are omitted. The non-linearity of the curves is due
to second order effects, which are not expected for pure Gaussian beams.

Figure 54:  Estimated drift of the beam separation in the crossing plane
$\Delta x^{\prime}$ in the (left) April and (right) July VDM sessions. The
drift is estimated only for each nominally head-on step.

The fast component of the beam separation drift is more difficult to estimate.
The presence of a crossing angle enables an estimation of the drift in the
beam separation in the crossing plane $\Delta x^{\prime}$. This is possible
because of the correlation between the $\Delta x^{\prime}$ and the $z$
position of the luminous region as seen in Eq. (19). While the latter is
strictly true only for Gaussian beams, the exact form of the function
$\xi_{lz}(\Delta x^{\prime})$ can be estimated by fitting the measured
$\xi_{lz}$ as function of $\Delta x^{\prime}$ with a smooth function. The data
and the estimated dependence are shown in Fig. 53. The deviations of the data
points $\xi_{lz}$ from the curve give an estimate of the beam drift. This
approach is only reliable when the reference curve is obtained by averaging
enough measurements. Therefore, the drift is estimated only for the three
nominally head-on steps for each scan, as shown in Fig. 54.

Figure 55:  Two example simulations of the beam separation drift in the $x$
coordinate for the July VDM session. The separation drift is modelled with a
Brownian motion that is constrained to the measured slow component of the
drift (smooth black line) at the measurement points (crosses).

Figure 56:  Cross-section bias caused by beam orbit drifts in the (left) April
and (right) July VDM sessions. The baseline cross-section for each pair is
shown with a solid circle and the average calibration cross-section (using the
reference pairs) is shown with a horizontal dashed line. The average biased
cross-section obtained with simulation of the beam drifts is shown with open
circles. The error bars indicate the RMS of the biased cross-section of 400
independent simulations. The larger error bars for the offset scan pairs
indicate that they are more sensitive to random drifts.

To estimate the effect of the orbit drifts on the cross-section measurement a
simulation approach is used. The fast component of the drift in each
coordinate is modelled with a probabilistic process according to a Brownian
motion. The diffusion coefficient, which is the parameter of the model, is
estimated from the measured fast component of the drift in $\Delta
x^{\prime}$. The value is found to be $0.005\,\mathrm{\micro m^{2}s^{-1}}$ and
it is assumed to be equal for both coordinates and constant during a fill. The
separation drift model is constrained to the measured slow component of the
drift at the measurement points. Examples of the result of the simulation are
shown in Fig. 55. The systematic uncertainty is evaluated by analysing 400
statistically independent simulations for each scan session, where the
simulated drift is added to the nominal beam separation. The average bias on
the cross-section and its RMS are summarized on Fig. 56. It is seen that the
average bias, which is mainly due to the slow component of the drift, is
small. On the other hand, the uncertainty in the bias, which is represented by
the RMS of all simulations and is driven by the fast drift component, is
sizeable.

Since the fast drifts are modelled to be independent for each scan pair, the
uncertainty on the average cross-section is reduced. The uncertainty due to
the slow component is assumed to be correlated, thus the estimates for
individual scan pairs are averaged. The two uncertainties are added in
quadrature to obtain a total uncertainty of 0.54% and 0.21% in April and July,
respectively. The non-reproducibility can be estimated directly by the
deviations from the average of the cross-section from individual scan pairs.
The maximum deviations for reference scan pairs observed in April and July are
$0.03\%$ and $0.30\%$, respectively. Assuming that the deviations are mainly
due to the drift, and in order to avoid double counting, only the larger value
of the drift estimate and the maximum deviation from the average is taken as a
systematic uncertainty.

It is seen in Fig. 56 that the scan pairs with an offset working point are
more sensitive to random beam drifts, which can be explained by the large
derivative of the VDM profile at non-zero beam separation. This is the main
reason for excluding offset scans from the cross-section measurement.
Moreover, since scan pair number five in July had a working point offset by
about $30{\,\micro\rm m}$ in $x$, it is also more sensitive to beam drifts.
Therefore, it is not included in the determination of the central value of the
cross-section. Considering the reference scan pairs (one to four) in July, the
uncertainties of the drift bias are too large compared to the observed
fluctuations. This suggests that the value used for the simulation parameter,
on which these uncertainties depend directly, is overestimated or the employed
model does not describe well the beam drifts. Therefore, the higher cross-
section measured from scan pair number five (see Fig. 40) cannot be explained
by random beam drifts and the deviation from the average cross-section (using
pairs one to four) of $0.8\%$ is taken as a systematic uncertainty to account
for a potential unknown source of non-reproducibility. The latter systematic
uncertainty is assigned to all $pp$ calibrations and is considered to be
correlated among calibrations.

The effects of the applied corrections are summarized in Table 11. The values
for the other energies will be discussed below (Sec. 7.10).

Table 11:  Effect of corrections on the result of the VDM scan calibrations at $\sqrt{s_{\mbox{\tiny{\it NN}}}}$ values of 8, 7 and 5$\rm\,TeV$ (in %). To obtain the effects, each correction is excluded, the fits are redone and the result is compared with the baseline.  | 8$\mathrm{\,Te\kern-1.00006ptV}$ | 7$\mathrm{\,Te\kern-1.00006ptV}$ | 5$\mathrm{\,Te\kern-1.00006ptV}$
---|---|---|---
Source | April | July |  |  $p$Pb | Pb$p$
Ghost charge | $+0.93$ | $+0.68$ | $+1.59$ | $+0.80$ | $+0.97$
Satellites charge | $+0.85$ | $+0.43$ | $+0.78$ | $+0.19$ | $+0.19$
Efficiency of the observable | $+0.32$ | $-0.03$ | $-0.04$ | $+0.08$ | $+0.11$
Length scale | $-1.39$ | $+0.39$ | $+0.06$ | $-2.14$ | $-1.03$
Dynamic $\beta$ | $-0.39$ | $-0.37$ | $-0.35$ | $-0.05$ | $-0.05$
Beam-beam deflection | $+1.12$ | $+1.12$ | $+1.00$ | $+0.15$ | $+0.16$
Linear correlation | $+0.30$ | n.a. | n.a. | n.a. | n.a.

###  7.9 Results

The reference cross-section for $pp$ collisions at
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ for the _Track_ observable is
determined by computing a weighted average of the results from the
calibrations in the April and July fills. It is assumed that most of the
systematic uncertainties are fully correlated in order to avoid
underestimating the uncertainty on the combined result. The individual
calibration results and the average reference cross-section are shown in Table
12. A list of all uncertainties for the July and April calibrations is
provided in Table 13 along with estimates of their correlation. The values for
the other energies are discussed below.

Table 12:  Individual calibration results and average of the $8\rm\,TeV$ $pp$ VDM scan sessions. The part of the relative uncertainty that is correlated between the calibrations is shown in the fourth column. The weights used to obtain the average are given in the last column.  Session | $\sigma_{\mathrm{Track}}$ (mb) | Relative | Correlated | Weight
---|---|---|---|---
uncertainty | part
April 2012 | $60.70\pm 0.92$ | $1.52\%$ | $1.34\%$ | 0.53
July 2012 | $60.54\pm 0.93$ | $1.54\%$ | $1.51\%$ | 0.47
Average | $60.62\pm 0.89$ | $1.47\%$ |  |
Table 13:  Relative systematic uncertainties on the reference cross-section for the VDM calibrations at $\sqrt{s_{\mbox{\tiny{\it NN}}}}$ values of 8, 7 and 5$\mathrm{\,Te\kern-1.00006ptV}$ (in %). The uncertainties are divided into groups affecting the description of the VDM profile (Secs. 7.5–7.8), the measurement of the rate (Sec. 7.4) and the bunch population product measurement (Sec. 3). The fourth (eight) column indicates whether the uncertainties are correlated between the two $pp$ calibrations at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ (the $p$Pb and Pb$p$ calibrations). Empty cells indicate that the corresponding source is not applicable or the uncertainty is negligible. For the $p$Pb and Pb$p$ calibrations, the uncertainties due to scan variation and drift, and non-reproducibility cannot be estimated separately as only two VDM scan pairs per calibration were performed.  | 8$\mathrm{\,Te\kern-1.00006ptV}$ | 7$\mathrm{\,Te\kern-1.00006ptV}$ | 5$\mathrm{\,Te\kern-1.00006ptV}$
---|---|---|---
Source | April | July | Corr. |  |  $p$Pb | Pb$p$ | Corr.
VDM profile description
Fit model | 0.50 | 0.50 | yes | 0.50 | 1.31 | 1.34 | yes
Fit bias | 0.20 | 0.20 | yes | 0.20 |  |  |
Linear correlation | 0.15 |  | no |  |  |  |
Parameter assumptions | 0.60 | 0.90 | yes | 0.90 |  |  |
Constraints from BGI | 0.30 | 0.30 | yes | 0.80 |  |  |
Length scale | 0.50 | 0.50 | yes | 0.50 | 1.10 | 1.10 | no
VELO transverse scale | 0.05 | 0.05 | yes | 0.05 | 0.05 | 0.05 | yes
Beam-beam effects | 0.27 | 0.30 | yes | 0.29 | 0.06 | 0.05 | yes
Scan variation and drift | 0.54 | 0.30 | no | 0.20 | 0.67 | 1.31 | no
Non-reproducibility | 0.80 | 0.80 | yes | 0.80 |  |  |
(fifth scan pair in July) |  |  |  |  |  |  |
Statistical | 0.06 | 0.05 | no | 0.11 | 0.26 | 0.25 | no
Rate measurement
Beam-gas background | 0.03 | 0.07 | yes | 0.08 | 0.34 | 0.21 | yes
Beam-beam background | 0.09 | 0.18 | yes | 0.09 | 0.65 | 0.71 | yes
Efficiency of rate observable | 0.13 | 0.07 | yes | 0.17 | 0.04 | 0.06 | yes
Bunch population uncertainties
DCCT population product | 0.22 | 0.24 | yes | 0.24 | 0.31 | 0.34 | no
FBCT offset | 0.04 | 0.06 | yes | 0.01 | 0.21 | 0.09 | no
BPTX cross-check | 0.13 | 0.06 | yes |  | 0.14 | 0.14 | no
Ghost charge | 0.05 | 0.03 | yes | 0.07 | 0.14 | 0.19 | no
Satellites charge | 0.02 | 0.03 | yes | 0.25 | 0.07 | 0.09 | no
No satellite measurements | 0.43 |  | no |  |  |  |
Total | 1.52 | 1.54 |  | 1.71 | 2.05 | 2.36 |

###  7.10 Summary of other van der Meer scan calibrations

The VDM analysis presented in this paper focuses on the calibration of the
reference cross-section at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ ($pp$).
Comparable results are obtained with the VDM method for the 2011 $pp$
calibration at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and the 2013 $p$Pb
and Pb$p$ calibrations at $\sqrt{s_{\mbox{\tiny{\it
NN}}}}=5\mathrm{\,Te\kern-1.00006ptV}$.

A VDM scan session was performed in fill 2234 for the calibration of the
reference cross-section at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ ($pp$).
The operational procedure and the relevant trigger configuration were very
similar to the one employed in 2012. The fill conditions and the scan
parameters are listed in Tables 1 and 8. Three symmetric $x$-$y$ scan pairs
were performed. The average decay time of the bunch population product amounts
to 70 hours and the luminosity drop caused by emittance growth is negligible.

The VDM analysis at 7$\mathrm{\,Te\kern-1.00006ptV}$ is performed in exactly
the same manner as for the 8$\mathrm{\,Te\kern-1.00006ptV}$ calibrations,
which are discussed in detail. Given the similarity of the conditions, the
systematic uncertainties on the fit model, the non-reproducibility and the
length scale calibration are directly translated from the 2012 calibration.
The corrections and the systematic uncertainty due to beam-beam effects are
found to be similar, owing to the similar beam energy and bunch intensities.
The dominating systematic uncertainties are related to the constraints on the
beam factorizability as obtained from the BGI analysis (Sec. 6.7). By using
these constraints instead of making parameter assumptions (see Sec. 7.3), the
measurement is shifted by 0.9%. The systematic uncertainty arising from the
uncertainty in the central values of the constraints is found to be 0.8% and
it is fully correlated with the BGI result.

Individual calibrations of the $p$Pb and Pb$p$ reference cross-sections were
performed in separate fills (3505, 3542). Due to the limited time of the ion
runs, normal fills for physics data taking were used. Therefore, the filling
schemes (see Table 1) were not optimized as for dedicated luminosity
calibration fills. The number of colliding bunches was 38, instead of usually
16 in dedicated fills. Moreover, the peak $\mu_{\mathrm{vis}}$ value was very
low, $<0.02$. In order to decrease statistical uncertainties, the random
trigger rate was doubled to 45${\rm\,kHz}$ during the $p$Pb calibration. The
statistical uncertainties on the cross-section measurement are significantly
larger compared to $pp$ calibrations and amount to $0.25\%$.

The statistical uncertainty on the cross-section measurement per colliding
bunch pair and per scan pair is about 2%. Therefore, a systematic structure in
the fit residuals, which leads to a measurement bias of that order, may not be
revealed by the $\chi^{2}/\mathrm{ndf}$ values of individual fits. The fit
quality of a model is estimated from the sum of the residuals of all bunch
pairs as function of beam separation. The aggregated residuals show a
statistically significant structure if a pure Gaussian model is employed. It
is found that two factorizable empirical models (a double Gaussian function
with a negative weight for the narrow component and a Gaussian function
multiplied by an even fourth order polynomial) fit the VDM scan profiles well
and give similar cross-section values. In addition, two non-factorizable
empirical models for the two-dimensional luminosity profiles have similar fit
quality. The first non-factorizable model is a sum of two two-dimensional
Gaussian functions with a negative weight for the narrow term. The second non-
factorizable model is a $xy$ rotationally symmetric Gaussian function
multiplied by an even fourth order polynomial and scaled by the $x$ and $y$
profile widths. The two non-factorizable models give similar cross-section
values, which are about 2% lower than the values from the factorizable models.
The VDM scans performed provide no information on the factorizability.
Therefore, the full range of obtained cross-section values is considered by
taking the central value at the middle and assigning half of the span as a fit
model uncertainty.

An independent length scale calibration was performed for both fills. The LHC
optics setup was almost identical in the two proton-lead beam configurations,
thus the calibration constants are expected to be identical. However, a
difference of about 1% is observed and is assigned as a systematic uncertainty
to both calibrations.

The uncertainty due to the beam orbit drift is estimated to be $0.67\%$ and
$0.88\%$ for the $p$Pb and the Pb$p$ calibration, respectively. The deviation
of the visible cross-section measurement from the average in repeated VDM
scans amounts to $0.23\%$ and $1.31\%$ for the $p$Pb and the Pb$p$
calibration, respectively. Following the procedure from Sec. 7.8, only the
larger value of the uncertainty due to orbit drift and the deviation from the
average is taken as a systematic uncertainty. Beam-beam effects have a small
impact on the $p$Pb and Pb$p$ calibrations owing to the low bunch intensities
of (1.3–1.7)$\times 10^{10}$ elementary charges. The low $\mu$ values
effectively increase the fraction of beam-gas and beam-beam induced
backgrounds. The corresponding uncertainties are evaluated as described in
Sec. 7.4 and amount to $~{}0.3\%$ and $~{}0.7\%$ for beam-gas and beam-beam
background respectively. It is assumed that there is no correlation between
the uncertainties of the $p$Pb and Pb$p$ calibrations when it is partial. This
is done to avoid underestimating the uncertainty on the ratio of the
luminosities of the two data samples, which enters into ratios of cross-
section measurements. A summary of all uncertainties is provided in Table 13.
The reference cross-section for the _Track_ observable is $2.126\pm
0.049\rm\,b$ and $2.120\pm 0.053\rm\,b$ for the $p$Pb and Pb$p$ mode,
respectively. The compatibility of the two results indicates that the _Track_
observable has similar efficiency for the two beam modes.

##  8 Summary and conclusion

Table 14:  Systematic uncertainties of the BGI and VDM methods for $pp$ interactions at $8\rm\,TeV$. The fourth column indicates whether the uncertainties are correlated between the two calibrations. All values are given in %.  Source | BGI | VDM | Correlated
---|---|---|---
Bunch population uncertainties (Sec. 3)
FBCT offset | 0.04 | 0.05 | yes
BPTX cross-check | n.a. | 0.09 | yes
DCCT population product | 0.22 | 0.23 | yes
Ghost charge | 0.02 | 0.04 | yes
Satellite charge | 0.06 | 0.02 | yes
Missing satellite measurements | n.a. | 0.23 | no
Rate measurement
Background subtraction | 0.20 | 0.14 | yes
Ratio of observables _Track_ to _Vertex_ | 0.20 | n.a. | no
Efficiency of rate observables | negl. | 0.09 | no
Fit model | 0.50 | yes
VELO transverse scale | 0.05 | yes
BGI specific (Sec. 6)
Beam-beam resolution | 0.93 |  | no
Beam-gas resolution | 0.55 |  | no
Detector alignment | 0.45 |  | no
Measurement spread | 0.54 |  | no
Bunch length | 0.05 |  | no
Reconstruction efficiency | 0.04 |  | no
Pressure gradient | 0.03 |  | no
VDM specific (Sec. 7)
Length scale |  | 0.50 | no
Beam-beam effects |  | 0.28 | no
Fit bias |  | 0.20 | no
Linear correlation |  | 0.08 | no
Parameter assumptions |  | 0.74 | no
Constraints from BGI |  | 0.30 | yes
Scan variation and drift |  | 0.32 | no
Non-reproducibility |  | 0.80 | no
Statistical |  | 0.04 | no
Uncorrelated | 1.31 | 1.32 |
Correlated | 0.59 | 0.65 |

Since some of the luminosity calibrations have been performed with both the
VDM and BGI method ($pp$ cross-section at
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ and
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$), the best result is obtained by
computing an average of the two methods, taking into account the correlation
between the systematic error sources. A summary of the final reference cross-
section results is presented in Table 15.

Table 15:  Results of the luminosity calibration measurements. The total uncertainty on the luminosity calibration (last column) is the sum in quadrature of the absolute calibration uncertainty (fourth column) and the relative calibration uncertainty (fifth column). The weights used to obtain the average absolute calibration at 8 and 7$\mathrm{\,Te\kern-1.00006ptV}$ ($pp$) are given in the third column. The part of the uncertainty that is correlated between VDM and BGI calibrations is shown in parentheses (fourth column).  Method | Absolute calibration | Relative calibration | Total
---|---|---|---
$\sigma_{\mathrm{vis}}$ (mb) | Weight | Uncertainty | uncertainty | uncertainty
$pp$ at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$
BGI | $60.62\pm 0.87$ | 0.50 | 1.43% (0.59%) |  |
VDM | $60.63\pm 0.89$ | 0.50 | 1.47% (0.65%) |  |
Average | $60.62\pm 0.68$ |  | 1.12% (0.00%) | 0.31% | 1.16%
$pp$ at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$
BGI | $63.00\pm 2.22$ | 0.13 | 3.52% (1.00%) |  |
VDM | $60.01\pm 1.03$ | 0.87 | 1.71% (1.00%) |  |
Average | $60.40\pm 0.99$ |  | 1.63% (0.00%) | 0.53% | 1.71%
$pp$ at $\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$
BGI | $52.7\phantom{0}\pm 1.2\phantom{0}$ |  | 2.20% (0.00%) | 0.25% | 2.21%
$p$Pb at $\sqrt{s_{\mbox{\tiny{\it NN}}}}=5\mathrm{\,Te\kern-1.00006ptV}$
VDM | $\phantom{.}2126\pm 49\phantom{.0}$ |  | 2.05% (0.00%) | 1.03% | 2.29%
Pb$p$ at $\sqrt{s_{\mbox{\tiny{\it NN}}}}=5\mathrm{\,Te\kern-1.00006ptV}$
VDM | $\phantom{.}2120\pm 53\phantom{.0}$ |  | 2.36% (0.00%) | 0.82% | 2.50%

The BGI and VDM calibrations of the visible $pp$ cross-section at
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ achieve very similar precision.
Therefore, for simplicity the two results are combined with equal weights; the
correlated components of the uncertainties are averaged linearly and the
others are averaged in quadrature. The different sources of uncertainty are
compared in Table 14, indicating also whether they are correlated or not. The
result of the combination is given in Table 15. If the uncertainty on the
propagation to physics data (0.31%, see Sec. 4) is included, the total
uncertainty on the luminosity is 1.16%. The latter uncertainty is valid if the
complete 2012 data set or a major part of it is used. In some cases, for small
partial sets, the uncertainty may be different.

A weighted average of the two $pp$ cross-section measurements at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ is computed, taking into account
that the VDM calibration is more precise. The correlated part of the
uncertainty due to the bunch population measurements and the factorizability
is estimated to be 1%. A weight of 0.87 is given to the VDM measurement based
on the uncorrelated part of the uncertainties. The final result given in Table
15 has a precision of 1.63%. If the uncertainty on the propagation to physics
data (0.53%, see Sec. 4) is also included, the total uncertainty on the
luminosity is 1.71%. As for the 2012 data set, the total uncertainty on the
luminosity is valid if a major part of the 2011 data set is used.

Figure 57:  Measurements of the visible cross-section for the _Vertex_
observable as function of centre-of-mass energy. The measurement at
2.76$\mathrm{\,Te\kern-1.00006ptV}$ [51] performed using the VDM method (solid
triangle) is corrected for the reduced efficiency due to the VELO being not
fully closed. The visible cross-section for proton-lead collisions at
5$\mathrm{\,Te\kern-1.00006ptV}$ is scaled by $A^{-2/3}$. A comparison is made
with the luminosity-independent measurements of the $pp$ inelastic cross-
section by the TOTEM collaboration [52, 8] at 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$, with direct measurements by the ALICE [53]
and the ATLAS [54] experiments, and with a measurement by ATLAS from elastic
$pp$ scattering [9]. The measurements from other experiments are scaled with
the LHCb efficiency for inelastic events $\eta_{\mathrm{\mbox{\scriptsize
Vertex,LHCb}}}$, obtained from simulation. The uncertainties of the direct
measurements from ALICE and ATLAS are dominated by the extrapolation of the
visible cross-section to the total inelastic cross-section and are not to be
compared with the uncertainties of the LHCb measurements. The tick marks
represent the uncertainty due to the luminosity calibration only. Data points
at the same centre-of-mass energy are displaced horizontally for clarity.

Measurements of the visible cross-section for the _Vertex_ observable as
function of centre-of-mass energy are shown in Fig. 57. The visible cross-
section for proton-lead collisions at 5$\mathrm{\,Te\kern-1.00006ptV}$ is
scaled by $A^{-2/3}$ ($A=208$ for lead). A comparison is made with the
luminosity-independent measurements of the $pp$ inelastic cross-section by the
TOTEM collaboration [52, 8] at 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$, and
direct measurements by the ALICE [53] and the ATLAS [54] experiments. The
measurements from other experiments are scaled with the LHCb efficiency for
inelastic events $\eta_{\mathrm{\mbox{\scriptsize Vertex,LHCb}}}$, which is
obtained from simulation with a negligible statistical uncertainty. The values
of $\eta_{\mathrm{\mbox{\scriptsize Vertex,LHCb}}}$ are 0.729, 0.768 and 0.758
for 2.76, 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$, respectively. A measurement
from CMS [55] is not shown as it does not include an extrapolation to the
total inelastic cross-section. No systematic uncertainties are included to
account for the $\eta_{\mathrm{\mbox{\scriptsize Vertex,LHCb}}}$ scaling, nor
for the scaling with $A^{-2/3}$.

Figure 57 also shows the result of a first luminosity calibration of the $pp$
cross-section at 2.76$\mathrm{\,Te\kern-1.00006ptV}$ which was performed using
the VDM method in 2011 [51]. During this data taking period, the VELO was
positioned with its sensitive area at a minimum distance of 13$\rm\,mm$ from
the beam instead of the nominal 8$\rm\,mm$. The corresponding drop in
efficiency is estimated to be 5% from simulation. Taking this difference into
account, as well as the unaccounted potential non-factorizability of the
beams, a good agreement is found when comparing to the more precise BGI
measurement from 2013.

An earlier measurement of the $pp$ visible cross-section based on the VDM and
BGI methods for the _Track_ observable at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ was reported in Ref. [10]. The data
available in the older publication were not sufficient to measure the
factorizability of the beams and complete factorizability was assumed. It was
shown here that neglecting effects of non-factorizability may cause an
underestimate of the visible cross-section at the few percent level.
Nevertheless, the value (58.8 $\pm$ 2.0$\rm\,mb$) is consistent with the
significantly more precise result reported here. The results reported here
supersede those of Ref. [10].

In conclusion, several luminosity calibration measurements were performed at
the LHC using the LHCb detector and two experimental methods, the VDM scan
method and the BGI method. Ghost charge fractions were also measured using
beam-gas interactions and the results are used in several luminosity
calibrations by other LHC experiments. The LHCb luminosity calibrations were
made for proton-proton collisions at three different centre-of-mass energies
(2.76, 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$) and for proton-lead collisions
at the equivalent nucleon-nucleon centre-of-mass energy of
5$\mathrm{\,Te\kern-1.00006ptV}$. The analysis strategies and the results of
the calibrations were presented in detail. Compared to the calibration
performed in 2010, an improvement by an order of magnitude was achieved in the
bunch population normalization, chiefly obtained by means of a thorough study
of the LHC beam current measuring devices. This achievement opened the way to
a global reduction of the systematic and statistical uncertainties for both
the VDM scan method and the BGI method. A controlled gas injection into the
LHC vacuum was employed to increase the beam-gas interaction rate by almost
two orders of magnitude and detailed systematic studies (including offset
scans and reproducibility checks) were conducted with the VDM scan method.
Modelling the non-factorizability in the transverse distribution of the bunch
particles is required and, if neglected, would have changed the calibration
results by up to 3%. In the case of proton-proton collisions at
8$\mathrm{\,Te\kern-1.00006ptV}$, a precision of 1.47% is obtained with the
VDM scan method and 1.43% with the BGI method. When combining the results, the
precision obtained on the reference visible cross-section is 1.12%, which
constitutes to date the most precise luminosity calibration at a bunched-beam
hadron collider. The precision of the calibrations for the other beams and
beam energies is close to 2%. The luminosity calibration results are used to
determine a reference cross-section, which is employed in the LHCb physics
data analysis to measure absolute cross-sections of various processes.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland);
INFN (Italy); FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA
(Romania); MinES and FANO (Russia); MinECo (Spain); SNSF and SER
(Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA). The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages on which we depend. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia). Individual groups or members have received support from EPLANET,
Marie Skłodowska-Curie Actions and ERC (European Union), Conseil général de
Haute-Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR
(Russia), XuntaGal and GENCAT (Spain), Royal Society and Royal Commission for
the Exhibition of 1851 (United Kingdom).

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LHCb collaboration

R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
Alvarez Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L.
An3, L. Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, C. Baesso60, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, V.
Battista39, A. Bay39, L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1,
S. Belogurov31, K. Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S.
Benson38, J. Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van
Beuzekom41, A. Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M.
Bjørnstad54, T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A.
Bondar34, N. Bondar30,38, W. Bonivento15,38, S. Borghi54, A. Borgia59, M.
Borsato7, T.J.V. Bowcock52, E. Bowen40, C. Bozzi16, T. Brambach9, J.
Bressieux39, D. Brett54, M. Britsch10, T. Britton59, J. Brodzicka54, N.H.
Brook46, H. Brown52, A. Bursche40, J. Buytaert38, S. Cadeddu15, R.
Calabrese16,f, M. Calvi20,k, M. Calvo Gomez36,p, P. Campana18,38, D. Campora
Perez38, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,38,j, A. Cardini15, L.
Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20, L. Castillo Garcia38,
M. Cattaneo38, Ch. Cauet9, R. Cenci23, M. Charles8, Ph. Charpentier38, M.
Chefdeville4, S. Chen54, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26,
K. Ciba38, X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38,
H.V. Cliff47, J. Closier38, V. Coco38, J. Cogan6, E. Cogneras5, L.
Cojocariu29, G. Collazuol22, P. Collins38, A. Comerma-Montells11, A.
Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G. Corti38, M. Corvo16,f,
I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz Torres60, S.
Cunliffe53, R. Currie53, C. D’Ambrosio38, J. Dalseno46, P. David8, P.N.Y.
David41, A. Davis57, K. De Bruyn41, S. De Capua54, M. De Cian11, J.M. De
Miranda1, L. De Paula2, W. De Silva57, P. De Simone18, C.-T. Dean51, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach55, O.
Deschamps5, F. Dettori38, A. Di Canto38, H. Dijkstra38, S. Donleavy52, F.
Dordei11, M. Dorigo39, A. Dosil Suárez37, D. Dossett48, A. Dovbnya43, K.
Dreimanis52, G. Dujany54, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A.
Dziurda26, A. Dzyuba30, S. Easo49,38, U. Egede53, V. Egorychev31, S.
Eidelman34, S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51, I. El
Rifai5, Ch. Elsasser40, S. Ely59, S. Esen11, H.-M. Evans47, T. Evans55, A.
Falabella14, C. Färber11, C. Farinelli41, N. Farley45, S. Farry52, RF Fay52,
D. Ferguson50, V. Fernandez Albor37, F. Ferreira Rodrigues1, M. Ferro-Luzzi38,
S. Filippov33, M. Fiore16,f, M. Fiorini16,f, M. Firlej27, C. Fitzpatrick39, T.
Fiutowski27, P. Fol53, M. Fontana10, F. Fontanelli19,j, R. Forty38, O.
Francisco2, M. Frank38, C. Frei38, M. Frosini17,g, J. Fu21,38, E. Furfaro24,l,
A. Gallas Torreira37, D. Galli14,d, S. Gallorini22,38, S. Gambetta19,j, M.
Gandelman2, P. Gandini59, Y. Gao3, J. García Pardiñas37, J. Garofoli59, J.
Garra Tico47, L. Garrido36, D. Gascon36, C. Gaspar38, R. Gauld55, L. Gavardi9,
A. Geraci21,v, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A.
Gianelle22, S. Gianì39, V. Gibson47, L. Giubega29, V.V. Gligorov38, C.
Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, C. Gotti20, M.
Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E. Graugés36,
G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P. Griffith45, L.
Grillo11, O. Grünberg62, B. Gui59, E. Gushchin33, Yu. Guz35,38, T. Gys38, C.
Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C. Haines47, S. Hall53, B.
Hamilton58, T. Hampson46, X. Han11, S. Hansmann-Menzemer11, N. Harnew55, S.T.
Harnew46, J. Harrison54, J. He38, T. Head38, V. Heijne41, K. Hennessy52, P.
Henrard5, L. Henry8, J.A. Hernando Morata37, E. van Herwijnen38, M. Heß62, A.
Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54, W. Hulsbergen41, P. Hunt55,
N. Hussain55, D. Hutchcroft52, D. Hynds51, M. Idzik27, P. Ilten56, R.
Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, P. Jaton39, A. Jawahery58,
F. Jing3, M. John55, D. Johnson38, C.R. Jones47, C. Joram38, B. Jost38, N.
Jurik59, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38, S.
Karodia51, M. Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C.
Khurewathanakul39, S. Klaver54, K. Klimaszewski28, O. Kochebina7, M. Kolpin11,
I. Komarov39, R.F. Koopman42, P. Koppenburg41,38, M. Korolev32, A.
Kozlinskiy41, L. Kravchuk33, K. Kreplin11, M. Kreps48, G. Krocker11, P.
Krokovny34, F. Kruse9, W. Kucewicz26,o, M. Kucharczyk20,26,k, V.
Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38,
G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, G. Lanfranchi18, C.
Langenbruch48, B. Langhans38, T. Latham48, C. Lazzeroni45, R. Le Gac6, J. van
Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32, J. Lefrançois7, S. Leo23, O.
Leroy6, T. Lesiak26, B. Leverington11, Y. Li3, T. Likhomanenko63, M. Liles52,
R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, S. Lohn38, I. Longstaff51,
J.H. Lopes2, N. Lopez-March39, P. Lowdon40, H. Lu3, D. Lucchesi22,r, H. Luo50,
A. Lupato22, E. Luppi16,f, O. Lupton55, F. Machefert7, I.V. Machikhiliyan31,
F. Maciuc29, O. Maev30, S. Malde55, A. Malinin63, G. Manca15,e, G.
Mancinelli6, A. Mapelli38, J. Maratas5, J.F. Marchand4, U. Marconi14, C. Marin
Benito36, P. Marino23,t, R. Märki39, J. Marks11, G. Martellotti25, A.
Martens8, A. Martín Sánchez7, M. Martinelli39, D. Martinez Santos42,38, F.
Martinez Vidal64, D. Martins Tostes2, A. Massafferri1, R. Matev38,w, Z.
Mathe38, C. Matteuzzi20, B. Maurin39, A. Mazurov45, M. McCann53, J.
McCarthy45, A. McNab54, R. McNulty12, B. McSkelly52, B. Meadows57, F. Meier9,
M. Meissner11, M. Merk41, D.A. Milanes8, M.-N. Minard4, N. Moggi14, J. Molina
Rodriguez60, S. Monteil5, M. Morandin22, P. Morawski27, A. Mordà6, M.J.
Morello23,t, J. Moron27, A.-B. Morris50, R. Mountain59, F. Muheim50, K.
Müller40, M. Mussini14, B. Muster39, P. Naik46, T. Nakada39, R. Nandakumar49,
I. Nasteva2, M. Needham50, N. Neri21, S. Neubert38, N. Neufeld38, M. Neuner11,
A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,q, M. Nicol7, V. Niess5, R.
Niet9, N. Nikitin32, T. Nikodem11, A. Novoselov35, D.P. O’Hanlon48, A.
Oblakowska-Mucha27,38, V. Obraztsov35, S. Oggero41, S. Ogilvy51, O.
Okhrimenko44, R. Oldeman15,e, C.J.G. Onderwater65, M. Orlandea29, J.M. Otalora
Goicochea2, P. Owen53, A. Oyanguren64, B.K. Pal59, A. Palano13,c, F.
Palombo21,u, M. Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51,
L.L. Pappalardo16,f, C. Parkes54, C.J. Parkinson9,45, G. Passaleva17, G.D.
Patel52, M. Patel53, C. Patrignani19,j, A. Pearce54, A. Pellegrino41, M. Pepe
Altarelli38, S. Perazzini14,d, P. Perret5, M. Perrin-Terrin6, L. Pescatore45,
E. Pesen66, G. Pessina20, K. Petridis53, A. Petrolini19,j, E. Picatoste
Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50,
M. Plo Casasus37, F. Polci8, A. Poluektov48,34, E. Polycarpo2, A. Popov35, D.
Popov10, B. Popovici29, C. Potterat2, E. Price46, J.D. Price52, J.
Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39,
G. Punzi23,s, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, F. Redi53, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S.
Ricciardi49, S. Richards46, M. Rihl38, K. Rinnert52, V. Rives Molina36, P.
Robbe7, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez Perez54, S. Roiser38, V.
Romanovsky35, A. Romero Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, H.
Ruiz36, P. Ruiz Valls64, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B.
Saitta15,e, V. Salustino Guimaraes2, C. Sanchez Mayordomo64, B. Sanmartin
Sedes37, R. Santacesaria25, C. Santamarina Rios37, E. Santovetti24,l, A.
Sarti18,m, C. Satriano25,n, A. Satta24, D.M. Saunders46, D. Savrina31,32, M.
Schiller42, H. Schindler38, M. Schlupp9, M. Schmelling10, B. Schmidt38, O.
Schneider39, A. Schopper38, M. Schubiger39, M.-H. Schune7, R. Schwemmer38, B.
Sciascia18, A. Sciubba25, A. Semennikov31, I. Sepp53, N. Serra40, J. Serrano6,
L. Sestini22, P. Seyfert11, M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30,
T. Shears52, L. Shekhtman34, V. Shevchenko63, A. Shires9, R. Silva Coutinho48,
G. Simi22, M. Sirendi47, N. Skidmore46, T. Skwarnicki59, N.A. Smith52, E.