Introduction

Overview

Teaching: 30 min
Exercises: 10 min

Questions

• What is luminosity?

• What is the difference between instantaneous and integrated luminosity?

• Why is knowing the luminosity important?

• How is luminosity measured?

Objectives

• Know what luminosity is and why it is important

• Know how luminosity is measured

References

You can find several papers with much more technical detail and several articles with additional (less formal) information in the References.

Instantanous Luminosity

In the context of the LHC, instantaneous luminosity, \(\mathcal{L}_{inst}\), corresponds to the number of “interactions” produced when “bunches” of protons are crossed. Roughly speaking, it corresponds to the “real-time rate of interactions/events/collisions”. During Run 2 of the LHC, groups of ~100 billion protons were crossed as often as 40 million times per second yielding an overall average of 34 interactions per crossing within the CMS detector.

More precisely, instantaneous luminosity quantifies the ability of particle accelerator to produce a certain number of interactions. It represents a proportionality factor between rate of interactions \(\left( \frac{dN}{dt} \right)\) and the cross-section (\(\sigma\)):

Thus, instantaneous luminosity is usually expressed in the cgs units of \(\mathrm{cm^{-2} s^{-1}}\). Units of “barns” are also used frequently, where \(1 \mathrm{b} = \mathrm{10^{-24} cm^{2}}\), thanks to two Purdue University physicists working on the Manhattan Project in 1942. As an example, let’s very approximately calculate the total Higgs Boson production rate at CMS:

1.1 Total Higgs boson production rate at CMS

• During May 2018, the LHC routinely delivered instantanous luminosities of \(\approx 2 \times 10^{34} \mathrm{cm^{-2} s^{-1}}\) \(\left( 0.02 \mathrm{pb^{-1} s^{-1}} \right)\) at CMS
• The total production cross section of Standard Model Higgs boson at \(\sqrt{s} = 13 \mathrm{TeV}\) can be slightly underestimated as \(\approx 50 \mathrm{pb}\) (see table 11.2 in the 2022 PDG)

What is the rate of Higgs production at CMS? Vote for the corect answer in the short lumi exercise Mattermost channel.

\[\frac{d\mathrm{N_{Higgs}}}{dt} = \mathcal{L}_{inst}^{\mathrm{peak}} \cdot \sigma_{\mathrm{Higgs}}^{\mathrm{total}}\]

Integrated Luminosity

Instantaneous luminosity is aggregated over a certain period of time to obtain integrated luminosity:

It is commonly used to quantify the “amount of data” delivered by the accelerator or recorded by the experiment. Units of inverse femtobars \(\mathrm{fb^{-1}}\) are frequently used in CMS.

To illustrate, we can very roughly estimate the total number of Higgs bosons produced during 24 hours at CMS:

1.2 Total Higgs bosons produced at CMS during 24 hours

• During Nov 2017, CMS recorded \(\approx 600 \mathrm{pb^{-1}}\) during a 24-hour period of stable beams
• The total production cross section of Standard Model Higgs boson at \(\sqrt{s} = 13 \mathrm{TeV}\) can be slightly underestimated as \(\approx 50 \mathrm{pb}\) (see table 11.2 in the 2022 PDG)

How many Higgs bosons can be produced at CMS during 24 hours? Vote for the corect answer in the short lumi exercise Mattermost channel.

\[\mathrm{N_{Higgs}} = \mathcal{L}_{int}^{\mathrm{24hr}} \cdot \sigma_{\mathrm{Higgs}}^{\mathrm{total}}\]

Importance of Luminosity

Along with the center of mass energy, instantanous luminosity is the most significant performance parameter for any particle accelerator. Real-time monitoring of instantaneous luminosity is critical for the accelerator to carry out beam tuning and collision optimization. It is also essential for the CMS trigger system in order to scale or throttle the data throughput.

Measurement of integrated luminosity is also incredibly crucial since its limited precision represents a contribution to the systematic uncertainty for most physics searches and measurements. The uncertainty in the integrated luminosity is often the dominant systematic uncertainty in EWK cross-section measurements. We can emphasize the impact of the integrated luminosity uncertainty by considering a relatively rare process:

1.3 Total Higgs bosons decaying to muon pairs at CMS during 2018

• During 2018, CMS recorded around \(\approx 60 \mathrm{fb^{-1}}\) of good-quality data with 2.5% uncertainty (see the CMS Lumi POG “quick summary table”)
• The total production cross section of Standard Model Higgs boson at \(\sqrt{s} = 13 \mathrm{TeV}\) can be slightly underestimated as \(\approx 50 \mathrm{pb}\) (see Table 11.2 in the 2022 PDG)
• The branching ratio for \(H \rightarrow \mu \mu\) can be approximated as \(\approx 2 \times 10^{-4}\) (see Table 11.3 in the 2022 PDG)

What is the minimum and maximum expected event yield given the uncertainty in integrated luminosity? Vote for the corect answer in the short lumi exercise Mattermost channel.

\[\mathrm{N}_{H \rightarrow \mu \mu}^{2018} = \left( \mathcal{L}_{int}^{2018} \pm \delta \right) \cdot \sigma_{\mathrm{Higgs}}^{\mathrm{total}} \cdot \mathcal{B}_{H \rightarrow \mu \mu}\]

Luminosity Measurement

CMS has two dedicated systems for measuring luminosity, both located \(z \approx \pm 1.8 \mathrm{m}\) from the interaction point and radius \(\approx 6 \mathrm{cm}\):

Fast Beam Condition Monitor (BCM1F)

C-shaped PCBs arranged into two rings at each side of CMS with double-pad silicon sensors

Real-time histogramming with 6.25 ns per-bin facilitates measurement of machine-induced background

Pixel Luminosity Telescope (PLT)

16 “telescopes” (8 per side of CMS) with three hybrid silicon pixel sensors per telescope

Fast cluster-counting signal (40 MHz) in addition to full pixel info readout

In addition, several sub-detectors are used for luminosity measurement, among them:

PCC (Pixel Cluster Counting)

Counts the mean number of pixel clusters in the most “stable” modules of the silicon pixel detector

Hadronic Forward (HF)

Steel absorber with quartz fibers to detect Cherenkov light histogrammed as function of bunch crossing

Which detector is more photogenic?

This is the most important poll!

https://indico.cern.ch/event/1239959/surveys/4059

Luminosity Calibration

The precise determination of integrated luminosity is particularly challenging at hadron colliders, in part due to the theoretical predictions (e.g. uncertainties in the parton distribution functions and precision of parton-level cross-section calculations) being generally less precise compared to \(e^{+} e^{−}\) colliders. A sub-detector can measure “relative” luminosity on an arbitrary scale based on the reported event rate. The determination of “absolute” luminosity involes re-scaling the measured event rate by a proportionality factor, \(\sigma_{vis}\), derived from the properties of the colliding beams. This scaling factor may be thought of as a way to account for the sub-detector’s particular acceptance and response.

At the LHC, the primary technique to determine the absolute luminosity scale is the van der Meer (vdM) scan method, based on dedicated beam-separation scans. The size and shape of the interaction region is measured by recording the relative interaction rates as a function of the transverse beam separation. After adopting several assumptions (e.g. transverse and longitudinal beam densities are Gaussian, density functions are factorizable into \(x\)- and \(y\)-dependent components, etc.), the visible cross-section can be expressed as

where \(\mu_{vis}^{\mathrm{max}}\) is the peak visible interaction rate, \(n_{1}\) and \(n_{2}\) are the numbers of particles in each of the two bunches, and \(\Sigma_{x}\) and \(\Sigma_{y}\) correspond to the effective beam overlap widths in each scan plane.

Corrections to vdM scan data

Several systematic effects can affect the measurement of \(\sigma_{vis}\). These represent a significant contribution to the final uncertainty in the measurement of integrated luminosity.

Orbit drift corrections

Potential bias from beam positions monitors (DOROS, Arc BPM) at the \(\mu \mathrm{m}\) scale

Beam-beam effects

EM interaction between colliding bunches (deflection & shape)

Length scale calibration

Possible differences in the absolute scale between the nominal beam separation (produced by the steering of the LHC magnets) and the actual separation

Transverse factorizability

Non-factorizability of \(x\) and \(y\) components measured and corrected with the beam-image method

Beam-imaging method: the distributions of reconstructed vertices during beam-imaging scans are used to obtain an image of the transverse bunch profiles integrated over the scanning direction

Other corrections

“Spurious” charges present outside the nominally filled bunches (ghosts👻 in empty bunch slots and out-of-time satellite🛰 charges adjacent to the main bunch)

Dominant uncertainties in the absolute luminosity scale (\(\sigma_{vis}\))

• beam position monitoring
• transverse factorizability
• beam-beam effects

Rate corrections under physics running conditions

Several corrections must be applied to luminometer rates to ensure that the final luminosity values are accurate.

Out-of-time pileup corrections

Most detectors have out-of-time contributions that do not arise from the main colliding bunch (spillover of electronic signals and real additional response from material activation)

Efficiency corrections

Radiation damage can affect the detector response by reducing efficiency, increasing noise, or both

Nonlinear response

Over- or under-counting as a function of instantaneous luminosity

Detector stability and linearity

Determined from comparisons between luminometers

Key Points

• Luminosity is a measure of how many collisions are delivered to and recorded by the detector.

• Instantaneous luminosity is usually expressed as the number of collisions per square centimeter per second.

• Integrated luminosity is the integral of instantaneous luminosity over time and is a measurement of data size. It is usually expressed in units of inverse cross section.

• Knowing the luminosity is important to determining and measuring accelerator and detector performance and operation. It is also an essential component for measuring cross sections and for setting limits on beyond-SM processes.

• Measurement of luminosity is done with several systems in the detector.