Luminosity Determination

In HERA-B several methods for luminosity determination are available [#!Lumi!#]. The algorithms used for luminosity determination of minimum-bias data taken in 2002/2003 are discussed in this section. The luminosity $L$ is the number of particles passing down the line per unit time, per unit area, and can be expressed as:

$$L= \frac{\mathrm{d}N}{\mathrm{d}\sigma},$$ (6.6)

where $\mathrm{d}N$ is the number of particles per unit time passing through area $\mathrm{d}\sigma$. At fixed target experiments the luminosity is proportinal to the number of beam particles hitting the target. The time integrated luminosity in case of HERA-B is defined as:

$$L= \frac { N_{proc} } { \sigma_{proc} } ,$$ (6.7)

where $N_{proc}$ is the number of interactions of a process of a specific type in a given time interval and $\sigma_{proc}$ is the process cross section. The cross section of inelastic processes $\sigma_{inel}$ dominates the total $pN$ cross section for HERA-B energies. Therefore one uses for the luminosity determination, the inelastic cross section as reference. Another advantage is that $\sigma_{inel}$ is known with a good accuracy and measured for a large variety of target materials and beam energies. Taking into account the HERA proton bunch structure and HERA-B setup of 2002/2003, the luminosity can be expressed as follows:

$$L= \frac { N_{BX}^{SLT}} { \sigma_{inel} } \lambda,$$ (6.8)

where $N_{BX}^{SLT}$ indicates how many times filled bunches crossed the target region and $\lambda$ is the mean number of interactions for filled bunches.

For the luminosity determination of the minimum bias data of 2002/2003 three different methods were used:

1. Hodoscope counters. For the interaction rate measurement four pairs of scintillators mounted in front of the ECAL are used, they are placed symmetrically around the proton beam. Hodoscopes provide a good linearity between rate and mean number of interactions but they have only a small acceptance.
2. ECAL energy sum method. The idea used in this method is that the mean total energy deposited in the ECAL is proportional to the average number of superimposed interactions.
3. Primary vertex counting method and counting of tracks from primary vertices. The algorithm is based on the assumption that the number of reconstructed primary vertices and the number of tracks assigned to the primaries scales linearly with the average number of superimposed interactions.

Hodoscope counters

The rate in the HERA-B setup 2002/2003 was measured by four pairs of scintillators, placed symmetrically around the beam pipe. Each counter has a geometrical acceptance equal to approximately 0.15%. The acceptance of these counters has been calibrated relative to a large acceptance hodoscope ($\approx$ 54% acceptance). This large acceptance hodoscope was temporary installed inside the magnet. Interaction rate measured by the hodoscope counters can be expressed as follows

$$\lambda = \frac{ R_{\rm {HOD}}\cdot \varepsilon}{R_{\rm {BX}}},$$ (6.9)

where $R_{\rm {HOD}}$ is the rate measured by the hodoscopes, $\varepsilon$ is an acceptance correction factor, $R_{\rm {BX}}=180 R_0/220$ is the rate of the non-empty bunches crossing the target region and $R_0=1/96$ ns is the bunch crossing rate. However, there are several reasons why additional sources for rate calculation are needed:

• The acceptance of the counters was measured in 1998. Later the large acceptance hodoscope was removed to install the tracking stations. Therefore the acceptance of the small counters has changed, since it is depending on the amount of material in front of them.
• Sensitivity of the hodoscopes to non-interaction related background (coasting beam).

ECAL Energy Sum

The idea behind this method is that average energy measured in the ECAL proportional to the mean number of superimposed interactions.

The average energy $E^{(N)}$ deposited by events with exactly $N$ interactions is

$$E^{(N)}= \frac {\sum\limits^{N_{total}}_{i=1} E_{i}I(n_i \approx N)} {\sum\limits^{N_{total}}_{i=1}I(n_i \approx N)},$$ (6.10)

where $N_{total}$ is the number of considered events, $E_{i}$ is the total energy deposited in the calorimeter per event and $n_i$ is the number of interactions per event. The mean energy can be expressed as

$$\bar E= \sum\limits^{ \infty }_{i=1} E^{(N)}P(N),$$ (6.11)

where $P(N)$ is the distribution function for the number of interactions $N$ with the mean value $\lambda$. Assuming that the energy scale linearly with the number of superimposed interactions, the mean number of interaction can be expressed as

$$\lambda = \frac{\bar E}{E^{(1)}}$$ (6.12)

The assumed linearity of the ECAL energy with respect to the number of interactions was checked by a Monte Carlo simulation and verified with experimental data. A typical distribution of the average energy dependence on the interaction rate for two wire materials carbon and titanium, obtained on data, is shown in Fig. 6.9.

Figure 6.9: Dependence of energy deposited in the ECAL as a function of the interaction rate for carbon and titanium wire obtained with the help of rate scans.

For the purpose of determining the luminosity, special runs have been taken, in which the interaction rate has been varied from the minimum till maximum value.

The energy of a single interaction can be determined from MC or data. A single interaction can be tagged in the zero-rate limit (at low rate the probability to have multiple interactions becomes negligible small), by requiring at least one cell with energy above threshold, such event is called ``tagged'' event. Taking into account the assumption that the number of interactions follows the Poisson statistics, the mean energy per tagged event can be defined as a function of the parameter $\lambda$

$$<E>_{tagged}=\frac{\lambda E^{(1)}}{1-e^{-\lambda \varepsilon(1)}},$$ (6.13)

where $\varepsilon(1)$ is the efficiency to tag an event (measured on MC), $E^{(1)}$ is the energy released with one interaction [#!Asom!#]. The energy $E^{(1)}$ is obtained from a fit to the mean energy of tagged events using function 6.13

Vertex Detector System based method

As an additional method the response of the Vertex Detector System is used. Assuming that the number of reconstructed tracks and vertices scales linearly with the number of interactions $N$, we can express $\lambda$ in a similar way as in the case of the ECAL energy sum method

$$\lambda = \frac{<n>_{\rm {tracks}}}{<n>_{\rm {tracks}}^{(1)}},$$ (6.14)

$<n>_{\rm {tracks}}$ is the average number of tracks assigned to vertices and $<n>_{\rm {tracks}}^{(1)}$ is the average number of tracks assigned to one interaction. In order to extract the average number of tracks for one interaction a similar function (6.13) as for the ECAL energy sum method is used to fit the distribution of track multiplicity versus interaction rate.

The obtained numbers ( $\varepsilon_{\rm {ECAL}}, \,\varepsilon_{\rm {vert}}, \, \varepsilon_{\rm {IA~trig}}$) are used to calculate the total number of filled bunches which crossed the target region and to correct the efficiencies of the applied cuts. The $N_{BX}^{SLT}$ for each of the methods can be expressed as follows

$$N_{BX}^{SLT}= \frac{N_{\rm {tape}}}{1-e^{-\lambda}-e^{-\lambda\varepsilon_{\rm {IA~trig}}}},$$ (6.15)

where $\lambda$ is the average number of interactions determined with the Hodoscope counters,

$$N_{BX}^{SLT}= \frac{N_{E>E_{\rm {thr}}}}{1-e^{-\lambda \varepsilo... ...^{-\lambda \varepsilon_{\rm {vert}}}-e^{-\lambda\varepsilon_{\rm {IA~trig}}} }.$$ (6.16)

The $\lambda$s are measured with random triggered data.

The numbers obtained with the different methods were compared for a set of runs for different wires. The results of the comparison are shown in Fig. 6.10 for Tungsten wire. The measurements obtained with the ``mean'' and ``Poisson'' method are in good agreement. The estimated systematic error of the luminosity measurements is $\approx$ 10%. The final luminosity numbers used in the analysis are listed in Appendix A.

Figure 6.10: Distribution of the average number of interactions measured with VDS and ECAL methods relative to the measurements obtained with Hodoscopes for the Tungsten wire [#!Asompriv!#].