Measurement of the front-end dead-time of the LHCb muon detector and evaluation of its contribution to the muon detection inefficiency

A method is described which allows to deduce the dead-time of the front-end electronics of the LHCb muon detector from a series of measurements performed at different luminosities at a bunch-crossing rate of 20 MHz. The measured values of the dead-time range from 70 ns to 100 ns. These results allow to estimate the performance of the muon detector at the future bunch-crossing rate of 40 MHz and at higher luminosity.


Introduction
The muon detector of the LHCb experiment [1 -3] is composed of 5 stations (M1−M5) placed along the beam axis. Each station is divided in 4 regions (R1−R4), with increasing distance from the beam pipe. Multiwire proportional chambers (MWPC) are used everywhere, except in the most irradiated region R1 of station M1 where triple-GEM [4,5] were adopted. The detector comprises 1380 chambers with 122112 readout channels. Each chamber is segmented in anode and /or cathode elements named pads. In the front-end (FE), each pad is read-out by a CARIOCA chip [6] which performs the signal amplification, shaping and discrimination. In the first data taking period (years 2009-2013) the LHCb experiment ran at a luminosity of up to 4 × 10 32 cm −2 s −1 and at a bunchcrossing (BC) rate R 0 = 20 MHz. To investigate the performance of the muon detector at the future BC rate of 40 MHz and at higher luminosity, the knowledge of the dead-time of the CARIOCA is crucial.
The present paper is divided in two parts. In the first part (Section 2) we describe the method adopted to deduce the dead-time of the CARIOCA (δ c ) from a measure of background rates (R * ) at different luminosities. These measurements were performed in dedicated runs with a non-standard data aquisition (DAQ).
In the second part of this paper (Section 3) the values of δ c obtained in Section 2 are used to evaluate the muon detection inefficiency due to the CARIOCA dead-time when the experiment runs in the standard data taking conditions and at a BC rate of 20 MHz and 40 MHz.

Measurement of the CARIOCA dead time with background events
To determine the CARIOCA dead-time, the behaviour of the FE electronics was measured in dedicated runs at √ s = 8 TeV, at a BC rate of 20 MHz and at five different luminosities: L 1 = 4 × 10 32 , L 2 = 5 × 10 32 , L 3 = 6 × 10 32 , L 4 = 8 × 10 32 and L 5 = 1 × 10 33 cm −2 s −1 . In these runs, background events were counted by free-running scalers placed in each FE board, downstream of the discriminator. These scalers are asynchronous with the LHC clock and can be controlled remotely. The dead-time of the scalers is negligible compared with that of the CARIOCA.
The measurements were performed on the readout channels of the most irradiated MWPCs belonging to stations M1 and M2. The counting rates of these channels are mainly due to low energy background particles. The contribution of the electronic noise to the measured rate was evaluated without beams and found to be negligible.

Dead-time with random particles
If the time distribution of the particles hitting a pad would not have any particular time structure, the counting rate (R * ) of a given readout channel would be given by: where δ c is the CARIOCA dead time and R part is the rate of hitting particles. If R * is measured at a single luminosity, the value of δ c cannot be deduced from Eq. 2.1 because R part is unknown. Two measurements (R * i and R * j ) performed at two different luminosities (L i and L j ) are necessary. For each of them Eq. 2.1 becomes: For each pad the ratio ρ i j , which can be evaluated from the experimental data, is defined [7]: Taking into account that R part is proportional to the luminosity (R For each pad of the detector, ρ i j and R * j can be measured and reported on a bi-dimensional scatter plot. According to Eq. 2.4, the points of this plot should be aligned with a slope equal to δ c (1−β i j ). Therefore the measured value of this slope allows to evaluate δ c .
For particles arriving at random times, the number of hits on a pad (N 0 ) in a time interval ∆t follows a Poisson distribution: and the interarrival times (τ) obey an exponential distribution: With bunched particles this interarrival distribution is modified and the distribution of the number of hitting particles in a given time interval is no longer given by Eq. 2.5.
In the next section the real experimental situation will be considered.

Dead-time with bunched particles
In the measurements performed at the LHC collider, R part has a bunched time structure which reflects the sequence of the bunch crossings. Long trains of consecutive BCs occuring every 25 ns (R 0 = 40 MHz) or 50 ns (R 0 = 20 MHz) are followed by empty intervals with no BCs. The overall duty cycle is ∼ 70 %. Moreover there is a finite probability that more than one particle generated in the same interaction will hit, near in time, the same pad, dependently on its size and position and independently of luminosity.
A relation similar to Eq. 2.1 can still be written but δ c must be replaced by an "effective deadtime" δ e f f : where R * and R part are the rates during the BC trains. The value of δ e f f depends on: • the CARIOCA dead-time δ c , • the repetition rate of the BCs (20 MHz or 40 MHz), • the probability distribution function (pdf) of the number of particles (N part ) arriving on the pads for each BC (the pdf changes with the luminosity), • the time of arrival of particles on the pad, • the time fluctuations of the chamber and FE response.
The method described in Sec. 2.1 can still be applied provided the dependence of δ e f f on the luminosity is taken into account. Eqs.2.2 becomes: By introducing these equations in ρ i j defined by Eq. 2.3, Eq. 2.4 is replaced by the relation: As described in the previous section, ρ i j and R * j are measured for each pad and their values are reported on a bi-dimensional scatter plot. According to Eq. 2.9 the slope of a linear fit to this plot is equal to δ e f f β i j , an expression which therefore is computable from experimental data. To evaluate δ c from this slope, the dependence of δ e f f on the luminosity and on δ c has been estimated by a Monte Carlo (MC) simulation.

Monte Carlo simulation of background events
The MC simulates the time of arrival of the pulses during a large number of consecutive BCs. If two events have a time distance lower than the CARIOCA dead-time δ c 1 , the second one is lost.
For each BC the pdf of the number of particles hitting a pad (N part ) and their experimental time distribution are considered.

Probability distribution function of the number of hitting particles
The number of background particles N part hitting a pad in a BC depends on the number of p-p interactions (n int ) in the BC and on the number of particles (n part ) hitting the pad for a single p-p interaction. At a given luminosity, n int follows a Poisson statistics with a mean value 2 µ: For a single p-p interaction, the number of particles (n part ) crossing a pad follows also a Poisson distribution with a mean value ω: The value of ω represents the pad occupancy per interaction. It is characteristic of each pad and depends on its size and on its position in the detector.
The ω values are spread over several orders of magnitude and a maximum value of ∼ 0.05 was measured on M2R1. For a single BC generating an arbitrary number of interactions, the distribution P 3 (N part ) of the number of particles crossing a given pad follows a "bi-Poisson" distribution which results from the convolution of P 1 (n int ) and P 2 (n part ): Because the two variables n int and n part are uncorrelated, the average value ( N part ) of P 3 (N part ) is equal to the product of the average values of P 1 (n int ) and P 2 (n part ): (2.13) and the background particle rate is: 14) 1 A gaussian fluctuation of ±9 ns (rms) around δ c was assumed for each event. Therefore δ c represents the average CARIOCA dead-time. This fluctuation was estimated by laboratory tests. 2 The luminosity L is proportional to µ. At L = 4 × 10 32 cm −2 s −1 and at a BC rate of 20 MHz, the value µ = 1.7 was assumed.
In the MC, N part was extracted for each BC according to the distribution given by Eq. 2.12. Note that this distribution, accounting for the probability that more particles generated in the same interaction will hit the same pad, implies that, in presence of a dead time, the inefficiency (R part − R * )/R part = δ e f f R * (see Eq. 2.7) will not go to zero for µ → 0, while R * will go to zero.
Therefore δ e f f , as defined in Eq. 2.7, will tend to infinity for µ → 0. This issue will be resumed later and discussed in Appendix A.

Time distribution of the background hits
The time distribution of the signals depends essentially on the time of flight of the detected particles and on the fluctuations of the response of the chamber and its readout electronics. This distribution was extracted from measurements performed in special low luminosity runs where events were acquired in a time gate of 125 ns around the triggered BC (instead of 25 ns adopted in the standard data taking) [2,3]. Different time distributions were measured in different detector zones.
In the MC, for each BC, the hits were distributed in time according to the spectrum of Fig

Experimental results
The counting rates of the pads belonging to the regions of station M1 and M2 equipped with MWPCs were measured in 1 s by the scalers of the CARIOCA chip. To take into account the duty cycle of LHC, the values of R * were obtained by attributing the scaler counts to a time interval of 0.7 s. The measurements were performed at five different luminosities. Only the couples of measurements performed in the same day were chosen for the analysis 4 . The data belonging to the five selected pairs of luminosities (L i , L j ) were represented on a scatter plot ρ i j versus R * j . As an example we show (Fig. 3a) the results of the measurements performed on all the pads of station M2, and using the pair of luminosities (L 1 , L 3 ). Each point corresponds to a pad. Cathode and anode pads are shown separately. The difference in the slope for cathode and anode readout is mainly due to the CARIOCA chip which is slightly different for positive and negative pulses. The width of the bands is compatible with the expected statistical fluctuations on the number of counted pulses. The fluctuations are larger in regions R3 and R4 (Fig. 3a) where the counting rate is lower. The points averaged in bins of R * 3 , are reported in Fig. 3b. If no dead time effect was present, the points of Fig. 3a and 3b should be aligned around the line ρ 13 = 1. The experimental points are instead distributed around a line with a non-zero slope which, according to Eq. 2.9, is equal to δ   Table 1, for the MWPCs of station M1 (regions R2, R3 and R4) and M2 (regions R1 and R2), and for cathode and anode readout separately.
For each region and readout, the δ c values obtained from different pairs of luminosity are significantly different. The dispersion of these values, reported as the first error in the last column of Table 1, represents the main contribution to the systematic error on δ c .
The second error is due to the uncertainties on the time distributions. To evaluate this contribution, the time distribution of the background pulses was moved inside the band shown in Fig. 1 and the maximum variation was taken. Also the fluctuation of the CARIOCA dead-time around its average value δ c was varied in the interval 9 ± 4 ns (see footnote 1). This last variation has a small effect on the results.  Region & readout C pad (pF) (L 1 , The results referring to the same pad type (anode or cathode) of a given station and region, but obtained with different pairs of luminosities are in a reasonably good agreement. The deadtime of the cathode readout are systematically larger than those of the anode readout. Considering separately the anode and cathode readout, some spread is still present between the results belonging to different stations and regions. This can be due to different detector geometries and therefore capacitances.
Once δ c is determined, the values of δ e f f can be evaluated (Fig.2b). Finally, Eqs. 2.7 and 2.14 allow to calculate, for each pad, the corresponding value of ω. As an example the dependence of the pad occupancy per interaction ω on R * is shown in Fig. 5 separately for anode and cathode pads of station M2, for two luminosities L 1 and L 3 .

Front-end inefficiency for muon detection due to CARIOCA dead-time
Once δ c is determined with the described method, the detection inefficiency for muon hits of each front-end 5 due to the CARIOCA dead-time can be evaluated. In the standard data taking the hits are acquired in a 25 ns gate, corresponding to the triggered BC. If one or more particles (muon or background particles) hit the pad in the 25 ns gate, and at least one survives to the dead-time generated in a preceeding BC the pad is efficient.
For a given pad the muon rate is negligible compared to the background rate so that the muon detection inefficiency is due to the dead-time generated by background hits. This inefficiency can be represented by an effective dead-time for muons (δ where R muon is the true muon rate while R * muon is the measured muon rate. The first member of Eq. 3.1 represents the FE muon detection efficiency (ε muon ), while in the second member R * is the rate of background particles counted in the pad during the BC trains.

Muon Monte Carlo simulation
To evaluate δ (muon) e f f in the standard data taking conditions, a dedicated MC was set which simulates a muon hit occurred in a triggered BC superimposed to a particle background having a rate R part . The fraction of lost muons was evaluated and δ (muon) e f f was calculated from Eq. 3.1. In the triggered BC the time of the muon hit was extracted according to the distribution [2] shown in Fig. 1 (dashed  line), while in all the BCs the background hits were extracted as described in Sec. 2.3.

Muon Monte Carlo results
The Muon MC was run for the BC rate of the LHCb Run1 (20 MHz) and for the future BC rate of 40 MHz. In Fig. 6a the (in)dependence of δ  (Fig. 6b).   (ns), calculated with the muon MC, are reported in the last two columns for a BC rate of 20 MHz and 40 MHz. The first error on δ (muon) e f f is due to the dispersion of δ c measured at the five different pairs of luminosity while the second error was estimated by taking into account the uncertaities on the time distributions of the background and muon hits (see text).

Region & readout
The muon detection efficiency of a pad is equal to 1 − δ (muon) e f f R * where R * is the background rate of the pad measured during the duty cycle (70 %) of LHC.

Muon hit inefficiency in the past and future running conditions
The results reported in Table 2 allow to assess the FE performance in the past runs and predict its behaviour at the high rates foreseen in the future high-luminosity upgrade conditions. For this purpose we express the muon detection efficiency (Eq. 3.1) of a pad as a function of the rate of background particles (R part ) hitting the pad: where R part is the particle rate on the pad during the duty cycle (70 %) of LHC and δ e f f is a function of δ c (Fig. 2b). Taking into account the values of δ c and δ (muon) e f f reported in Table 2, Eq. 3.2 allows to calculate the muon detection efficiency as a function of R part , for the seven considered regions and readout. This efficiency is usually reported as a function of the particle rate per cm 2 R part = 0.7 R part /A, where 0.7 R part is the particle rate on the pad averaged on the running time and A is the pad area reported in Table 2  In Fig. 7 the muon detection efficiency for the seven considered regions and readout is reported as a function of R part . In Fig. 7a the curve segments correspond to the effective R * intervals measured at the LHC pp centre-of-mass energy, √ s = 8 TeV with R 0 = 20 MHz and luminosity L = 4 × 10 32 cm −2 s −1 . In Fig. 7b the curve segments are calculated for R 0 = 40 MHz and L = 2 × 10 33 cm −2 s −1 and an overall factor 1.64 was applied to the expected rates to take into account the increase of √ s from 8 TeV to 14 TeV.
Referring to the conditions of Fig. 7b, the number of pads irradiated with R part MHz/cm 2 is reported in Fig. 8. The results presented in Fig. 7b and in Fig. 8 show that in the future running conditions the inefficiency due to dead-time would be a limiting factor to the operation of MWPCs belonging to all the regions of station M1 and to the inner region of station M2. Unacceptably poor FE efficiencies are expected in a relatively small zone of these regions, near the beam pipe, where the rate of background (and of good muons) is higher. This effect contributed to the decision [8] of removing station M1 in the upgraded detector for future high-luminosity runs at 14 TeV. Regarding the inner part of M2, the effect of adding an additional shielding around the beam pipe has been studied [8] and other interventions on the detector configuration are under investigation.

Conclusions
The dead-time δ c of the front-end electronics of stations M1 and M2 of the LHCb muon detector was determined from the experimental background rates measured at different luminosities by freerunning front-end scalers. The values obtained range between ∼ 70 ns and ∼ 100 ns, with the dead-time of the cathode pad readout being systematically larger than that of the anode pads. These results allow to determine the muon detection inefficiency of a front-end channel due to the deadtime, which is the largest contribution to the detector inefficiency in station M1 and in the inner regions of M2. The muon detection inefficiency was evaluated as a function of the background rate per unit area of the pad for the LHCb Run1 bunch-crossing rate of 20 MHz and for the future condition when LHCb will run at a bunch-crossing rate of 40 MHz, at √ s = 14 TeV and at a luminosity of 2×10 33 cm −2 s −1 . The large front-end inefficiencies expected in the future conditions contributed to the decision of eliminating the station M1 and inserting an additional shielding around the beam pipe upstream of the inner region of station M2.

A. Dead-time effect on background counting and muon detection
In the free-running front-end counters the hit of a particle generated in the current BC can be lost because of the dead-time generated by a particle belonging to a preceding BC or by a particle belonging to the current BC. When the luminosity is decreased, the time interval between two BCs with interacting protons increases so that the counting losses due to preceeding BCs become negligible when L → 0. This is not true for the counting losses due to particles belonging to the current BC. In fact when L → 0 the number of interactions in the considered BC is n int = 1 and the average number of particles hitting the pad is ω = 0. Therefore when L → 0 the fraction of lost particles R lost /R part = (R part − R * )/R part tends to a non-zero limit 7 : (δ e f f R * ) = 0 (A.1)  Figure 9. MC results on the ratio between the rate of lost particle and the rate of hitting particles as a function of µ for ω = 0.02; (a) refers to background particles counted with the free-running scalers (see text) while (b) refers to muons counted in a 25 ns triggered gate (standard data taking situation). Open (full) points correspond to δ c = 70 (100) ns.
As shown in Fig. 9a this effect is correctly predicted by the MC. On the other hand when L → 0 also R * → 0 so that the last equality in Eq. A.1 implies that δ e f f → ∞. This explains the behaviour of δ e f f shown in Fig. 2a.
In the standard data taking the situation is quite different: the muon hit is acquired during a triggered gate of 25 ns. If in this gate the muon hit is cancelled by the dead-time of a background hit generated and detected in the same BC, a hit is counted. As a consequence, the fraction of muon hits lost depends only on the events belonging to the preceeding BC and goes to 0 when L → 0. Therefore δ (muon) e f f results, as expected, to be independent of µ. As shown in Fig. 9b this effect is correctly predicted by the MC. 7 If ω 1, which is the present case, this limit is equal to ω/2.