Luminosity

Page maintainer: Caro

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Intro

• The Luminosity ${\mathcal {L}}$ is a measure for how intensely a reaction is taking place at a given moment. It depends on both the features of ingredient A and ingredient B, where A and B can be two colliding beams. In case of a fixed target experiment like Hermes, A is the lepton beam and B the gas target, and ${\mathcal {L}}$ can be written as:
  

${\mathcal {L}}={\frac {\mathrm {d} N_{A}}{\mathrm {d} t}}\cdot \rho _{B}\propto$ beam current $\times$ target density
where $\mathrm {d} N_{A}/\mathrm {d} t$ is the number of incident particles per time unit ($\propto$ beam current), and $\rho _{B}$ the areal density of the target.

• At Hermes, the luminosity is determined from the reaction rate $R_{\mathrm {ref} }\;\;[1/s]$ and the cross section in the luminosity monitor $\sigma _{\mathrm {ref} }\;\;[\mu \mathrm {barn} =10^{-34}m^{2}]$:
  

${\mathcal {L}}={\frac {R_{\mathrm {ref} }}{\sigma _{\mathrm {ref} }}}$.
The measured reaction is the elastic scattering of the beam leptons off the target gas shell electrons, Bhabha scattering ($e^{+}e^{-}$) in case of a positron beam (plus pair annihilation) and Møller scattering ($e^{-}e^{-}$) in case of an electron beam., by coincidently detecting the two leptons in the lumi monitor.
In every-day life, the analyzer uses only the reaction rate $R$ from the data stream (below called LumiRate), whereas the cross section $\sigma$ is obtained from Monte Carlo simulations (lumi constant provided by the lumi experts).

• The generic relation between measured cross section $\sigma _{\mathrm {meas} }$ of a certain process, the collected number of signal events $N$ and the integrated luminosity $L$ is:
  

$\sigma _{\mathrm {meas} }={\frac {N}{L}}$

Essentials

How do I correctly sum up lumi?

• The analyzer is mostly interested not in the luminosity rate (counts per time unit), but in the time ($t$) integrated luminosity $L$:
  

$L:=\int _{\Delta {\mathcal {T}}}\mathrm {d} t\;\epsilon (t)\;{\mathcal {L}}(t)$,
where the time-dependent efficiency factor $\epsilon (t)$ accounts for dead time effects of the detector.
Whenever the analyzer wants to set any count rate (DIS, SIDIS, exclusive) in relation to lumi (= normalization), the LumiRate has to be weighted with the
dead time correction factor = accepted/generated trigger rate (ranging from 0 to 1).
The reason is that LumiRate is a scaler rate which is always incremented, also if the Hermes spectrometer is dead for a short time.
In the experiment, the coincidence rate of the luminosity monitor is read out once per burst $j$ and is provided in the uDSTs:
${\mathcal {L}}_{j}$=g1Beam.rLumiRate $[1/s]$ and g1Beam.rLumiFitBstGai $[1/s]$.
Lumi integration means summing up all bursts $j$ of interest:
$L=\sum _{j}\Delta \tau _{j}\epsilon _{j}{\mathcal {L}}_{j}$, where
$\left.\Delta \tau _{j}\right.$=g1DAQ.rLength $[s]$ is the duration of the burst and
$\left.\epsilon _{j}\right.$=g1DAQ.rDeadCorr21 or g1DAQ.rDeadCorr, depending on wether the analyzer makes a trigger 21 cut or NOT.

• If you are running on target-polarized data, sum up lumi on uDST-counter-level in g1Beam. In 2006 and 2007, lumi is provided on burst level, even though for historical (back-compatibility) reasons the variable uDST-counter still exists. For target-polarized data, a burst is split up into 3 uDST-counter sub-units: 1. target spin into one direction; 2. target spin undefined; 3. target spin into the other direction.

• Remark: also the lumi monitor trigger has an efficiency, so to be 100% correct, one should use g1DAQ_rDeadCorr/TriggEffLumi. (NEEDREFERENCE: was the lumi trigger eff. on scaler page 1? Does anybody do this? NEEDEXPERT:Dominik)

• If you use the integrated lumi for normalization purposes of your physics signal, you may sum up only bursts after burstlevel data quality, i.e. after the burst passed the burst-level cuts of the bad burst list and possible additional burstcuts you perform for your specific analysis. In HANNA language, this means you sum up lumi somewhere in the end of user_burstinit().
  

• Side remark: naturally, you integrate your lumi before event level cuts. This is important to keep in mind when you count "lumi" in your Monte Carlo by summing up the number of generated events iEvGen. This must happen before you perform event level cuts in user_event().

• The inclusive analyses always summed up lumi separately for the top and bottom detector, even though the luminosity monitor is the only Hermes detector (apart from Recoil components) which is not constructed in the top/bottom topology. However, data quality is performed separately for the spectrometer halves and also detection efficiencies can be spectrometer half dependent.

What is LumiFit?

• Analyzers should always use g1Beam.rLumiFitBstGai whenever it is available, which is called LumiFit hereafter (NEVER use g1Beam.rLumiFit).

• LumiFit is only provided for target polarized data, which means not for high density and not for Recoil running.

• Purpose of LumiFit.
  

The cross section for Bhabha scattering is spin dependent. Therefore, if the target shell electrons are afflicted with a residual polarization, a slightly target (and beam) spin dependent LumiRate is introduced ("Bhabha asymmetry"), spoiling the measurement of the desired physics asymmetry. The procedure of smoothing LumiRate in order to obtain LumiFit cancels out this systematic spin dependency, besides decreasing statistical fluctuations.
Side remark: a special case are the tensor data of 2000, for which LumiFit was calculated spearately for vector and tensor target states.

Normalization and lumi constant

• The luminosity is one measure to normalize the physics signal (number of DIS, SIDIS, exclusive events, ...). An alternative method is the normalization to the number of DIS events.
  • A consistency check of the two methods has e.g. been performed in Falk Meissner's thesis, p.54ff
  • See example of DIS/lumi for some fills in 2000.
  • See also the wiki pages Cross section (data) and Normalizing MC.

• The determination of cross section asymmetries requires only the measurement of relative luminosities (because one divides the yield difference by its sum), whereas absolute luminosity measurements are needed for e.g. the $F_{2}$ analysis or cross sections.

• The ratio lumi/current is a measure for the target density. The reason is that LumiRate is proportional to the number of target shell electrons seen by the beam per unit area, whereas the beam current is not.
  • See example for some fills in 2005: one can see when the target shift crew opened the valve for high density running.
  • The Beam Life Time contribution by the Hermes target gas is calculated as: $\tau _{H}={\frac {\tau _{0}\tau _{G}}{\tau _{0}-\tau _{G}}}$ $[$hours$]$.
    • The higher the target density, the more the life time of the lepton beam $\left.\tau _{G}\right.$ $[$hours$]$ is decreased (dominant contribution: Coloumb scattering $\propto Z^{2}$ in the target cell).
    • $\left.\tau _{0}\right.$ is the beam life time before target gas injection. The beam has a finite life time also with empty Hermes target because of the remaining gas in the beampipe ($p_{0}$ term in the electron machine language) and because of synchrotron radiation and higher order mode heating of the beam ($p_{1}$ and higher terms).

• Lumi Constant. The lumi constant $C_{\mathrm {Lumi} }[\mathrm {mbarn} ^{-}1]$ is needed to transform the integrated LumiRate/LumiFit $L$ (which is dimensionless) into an absolute integrated luminosity which has the unit "per area" (inverse picobarn):
  

Absolute Lumi $[\mathrm {picobarn} ^{-}1]$ = $L\cdot C_{\mathrm {Lumi} }\cdot {\frac {\mathrm {\#nucleons} }{\mathrm {\#electrons} }}$.

• • The lumi constant comprises the acceptance of the lumi monitor. It is obtained in Monte Carlo simulations. ** As it depends also on the beam parameters (positions and slopes) and charge, it is provided separately for each data taking year.
  • The lumi scans which were performed every now and then during data taking were dedicated to support the determination of the lumi constant.
  • The ratio #nucleons/#electrons takes into account that the nucleon density is growing faster in the nucleus than the shell electron density.
  • The lumi constants for 2006 and 2007 were updated in October 2011! Further down in the same mail thread you can also find Elke's mail with the beam offsets, Adel's plot with lumi constant versus shift of the center of the target distribution along the z axis, and Vitaly's study regarding normalization of physics yield.

When does our lumi increase?

• When the target density is increased because the shift crew opens the dosing valve more
• When HERA runs with electrons instead with positrons, as the Møller ($e^{-}e^{-}$) cross section is 50% higher than the Bhabha ($e^{+}e^{-}$) cross section
• When HERA injects higher currents the next fill
• When heavier gas targets are used (then nucleons can be packed denser)

More Info

Hardware

• Lumi Monitor homepage
• Internal note 01-009: Acceptance Corrections for the Hermes luminosity monitor
• Lumi NIM paper 2001 (including systematics for Run I)
• Lumi onsite tutorial May 2004

Lumi Constants

• Official lumi constants over the years
• Integrated luminosities for each year
• Internal note 02-013: Lumi Constants for Run I

Theses

• Thomas Benisch, Polarisierte Bhabha-Streuung und Luminositaetsmessung im HERMES-Experiment (PhD thesis, in German)
  • Includes a description about to account for accidental coincidences for 1996 and 1997 polarized running (p.74):

LumiRate(corrected) = LumiRate(measured) - LumiRate(accidental), with LumiRate(accidental) = LumiRate(left)*LumiRate(right)*80ns

• Dominik Gabbert, Determination of the Structure Function F2 at Hermes
  • Lumi measurement: section 4.8