Precision Luminosity of LHC Proton-Proton Collisions at 13 TeV Using Hit-Counting with TPX Pixel Devices

TL;DR

This paper demonstrates that a network of Timepix pixel detectors can measure LHC proton-proton collision luminosity at 13 TeV with high precision and stability, using hit-counting techniques during 2015 data-taking.

Contribution

It introduces a stand-alone TPX pixel detector network for precise luminosity measurement and analyzes its performance during LHC 13 TeV collisions.

Findings

Achieved 0.1% short-term precision in 10 s intervals

Long-term stability below 0.5% over data-taking period

Validated the capability to monitor luminosity reduction accurately

Abstract

A network of Timepix (TPX) devices installed in the ATLAS cavern measures the LHC luminosity as a function of time as a stand-alone system. The data were recorded from 13 TeV proton-proton collisions in 2015. Using two TPX devices, the number of hits created by particles passing the pixel matrices was counted. A van der Meer scan of the LHC beams was analysed using bunch-integrated luminosity averages over the different bunch profiles for an approximate absolute luminosity normalization. It is demonstrated that the TPX network has the capability to measure the reduction of LHC luminosity with precision. Comparative studies were performed among four sensors (two sensors in each TPX device) and the relative short-term precision of the luminosity measurement was determined to be 0.1% for 10 s time intervals. The internal long-term time stability of the measurements was below 0.5% for the data-taking period.

Tables

Table 1. TABLE I: TPX device locations with respect to the interaction point. Z 𝑍 Z is the longitudinal distance from the interaction point and R 𝑅 R is the distance from the beam axis. Their uncertainty is about 10 mm.

Device	$Z$	$R$	TPX clusters per unit sensor area and
	(mm)	(mm)	per unit luminosity $\left({\mathrm{cm}}^{-2}/{\mathrm{nb}}^{-1}\right)$
			Layer-1	Layer-2
TPX02	3540	1115	104	123
TPX12	-3540	1146	97	113

Table 2. TABLE II: Van der Meer (vdM) scan results for 2015 data. The scan was performed on 24-25 August (LHC fill 4266) and the first horizontal and vertical scans were used. The fit results for the first bunch-averaged horizontal Σ x subscript Σ x \Sigma_{\rm x} and vertical Σ y subscript Σ y \Sigma_{\rm y} convoluted beam sizes are given, as well as the specific luminosity. The hit rate at the peak is averaged over the horizontal and vertical scans.

TPX	Layer	${\Sigma }_x$	${\Sigma }_y$	$L_{\mathrm{specific}}$	$N_{\mathrm{peak}}$
		($\text{micron}$)	($\text{micron}$)	($\mu b^{-1}{s}^{-1}/{10}^{25}$)	($\mathrm{hits}/s$)
02	1	130.2	117.7	116.8	1495
02	2	130.2	118.6	115.9	2386
12	1	128.7	118.1	117.8	1386
12	2	128.3	118.7	117.5	2298

Table 3. TABLE III: Normalization ( 1 / n f 1 subscript 𝑛 f 1/n_{\rm f} ) to convert hit rates to luminosities. The larger values are for the thicker sensor layer.

Device	Layer	$1/n_f$
		(hits/$\mu b^{-1}$)
TPX02	1	544.8
TPX02	2	876.1
TPX12	1	500.9
TPX12	2	832.2

Table 4. TABLE IV: Parameters of the LHC proton-proton collisions for the 2015 fills analysed with TPX02 and TPX12 in the fitting of LHC luminosity reduction curve. L 0 subscript 𝐿 0 L_{0} is the luminosity and μ 0 subscript 𝜇 0 \mu_{0} is the average number of interactions per bunch crossing at the start of the LHC fill. The λ bb 0 superscript subscript 𝜆 bb 0 \lambda_{\rm bb}^{0} is given for the start of the LHC fill, and λ bb 1 superscript subscript 𝜆 bb 1 \lambda_{\rm bb}^{1} is given at the start time of the fit for the first time period.

Date	Fill	Start time	$k{N}_0$	$L_0$	${\mu }_0$	Colliding	${\lambda }_{\mathrm{bb}}^0$	${\lambda }_{\mathrm{bb}}^1$
in 2015		(unix time)	$\left({10}^{11}\right)$	$\left(\mu b^{-1}{s}^{-1}\right)$		bunches	$\left({\mathrm{day}}^{-1}\right)$	$\left({\mathrm{day}}^{-1}\right)$
29 Sep.	4440	1443525360	1650	3360	24.8	1453	0.2815	0.1833
2-3 Oct.	4449	1443785520	1607	3240	26.0	1453	0.2787	0.1615
6-7 Oct.	4467	1444149360	1729	3700	24.1	1596	0.2958	0.1873
9-11 Oct.	4479	1444427640	1950	4240	24.6	1813	0.3006	0.2194
30-31 Oct.	4557	1446221040	2388	4450	24.7	2232	0.2576	0.1840
31 Oct.-1 Nov.	4560	1446311100	2610	5020	24.5	2232	0.2659	0.1859

Table 5. TABLE V: Precision of TPX02 and TPX12 luminosity measurements for the average of layer-1 and layer-2. The precision is given for 1 frame and for 10, 20, 30 and 40 frames combined. The width of the Gaussian fit σ ( data − fit ) / fit subscript 𝜎 data fit fit \sigma_{\rm(data-fit)/fit} gives the precision of the measurement and σ pull subscript 𝜎 pull \sigma_{\rm pull} gives the width of the fit of the pull distribution with statistical uncertainties only. LHC fill 4449.

Frames	${\sigma }_{\left(\mathrm{data}-\mathrm{fit}\right)/\mathrm{fit}}$ (%)				${\sigma }_{\mathrm{pull}}$
used	TPX02		TPX12		TPX02		TPX12
	Layer-1	Layer-2	Layer-1	Layer-2	Layer-1	Layer-2	Layer-1	Layer-2
1	0.36	0.35	0.37	0.36	1.27	1.58	1.28	1.57
10	0.13	0.13	0.15	0.13	1.47	1.87	1.61	1.82
20	0.11	0.10	0.13	0.11	1.71	2.04	1.96	2.12
30	0.10	0.09	0.11	0.10	1.96	2.18	2.17	2.40
40	0.09	0.08	0.11	0.09	2.13	2.39	2.24	2.56

Table 6. TABLE VI: Precision of TPX02 and TPX12 luminosity measurements with respect to the fitted curve for the average of layer-1 and layer-2. The precision is given for 1 frame and for 10, 20, 30 and 40 frames combined. The width of the Gaussian fit σ ( data − fit ) / fit subscript 𝜎 data fit fit \sigma_{\rm(data-fit)/fit} gives the precision of the measurement and σ pull subscript 𝜎 pull \sigma_{\rm pull} gives the width of the fit of the pull distribution with statistical uncertainties only. LHC fill 4449.

Frames	${\sigma }_{\left(\mathrm{data}-\mathrm{fit}\right)/\mathrm{fit}}$    (%)		${\sigma }_{\mathrm{pull}}$
used	TPX02	TPX12	TPX02	TPX12
1	0.30	0.30	1.50	1.57
10	0.11	0.12	1.76	1.94
20	0.09	0.10	1.96	2.36
30	0.08	0.09	2.21	2.56
40	0.08	0.09	2.35	2.88

Table 7. TABLE VII: Precision of TPX02 and TPX12 luminosity measurements between layer-1 and layer-2. The precision is given for 1 frame and for 10, 20, 30 and 40 frames combined. The width of the Gaussian fit σ 2 ​ ( μ 1 − μ 2 ) / ( μ 1 + μ 2 ) subscript 𝜎 2 subscript 𝜇 1 subscript 𝜇 2 subscript 𝜇 1 subscript 𝜇 2 \sigma_{\rm 2(\mu_{1}-\mu_{2})/(\mu_{1}+\mu_{2})} gives the precision of the measurement and σ pull subscript 𝜎 pull \sigma_{\rm pull} gives the width of the fit of the pull distribution with statistical uncertainties only. LHC fill 4449.

Frames	${\sigma }_{2\left({\mu }_1-{\mu }_2\right)/\left({\mu }_1+{\mu }_2\right)}$ (%)		${\sigma }_{\mathrm{pull}}$
used	TPX02	TPX12	TPX02	TPX12
1	0.41	0.41	1.15	1.12
10	0.16	0.17	1.44	1.45
20	0.14	0.14	1.66	1.66
30	0.12	0.13	1.89	2.04
40	0.11	0.13	1.95	2.09

Table 8. TABLE VIII: Slope of time history of the luminosity ratio measured by layer-1 and layer-2 for TPX02 and TPX12. The slope values and the uncertainties are given per second and in percent per 100 days.

TPX	Slope	$\sigma$Slope	Slope	$\sigma$Slope
	(${10}^{-10}{s}^{-1}$)	(${10}^{-10}{s}^{-1}$)	(%/100d)	(%/100d)
02	$-7.06$	1.71	$-0.61$	0.15
12	$-4.51$	1.62	$-0.39$	0.14

Table 9. TABLE IX: Slope of time history of the luminosity ratio measured by TPX02 and TPX12 averaged over the data recorded by layer-1 and layer-2. The slope values and the uncertainties are given per second and in percent per 100 days.

TPX	Slope	$\sigma$Slope	Slope	$\sigma$Slope
	(${10}^{-10}{s}^{-1}$)	(${10}^{-10}{s}^{-1}$)	(%/100d)	(%/100d)
02/12	0.25	0.89	0.02	0.08

Equations

$$L_{\mathrm{TPX}}=N_{\mathrm{b}}{N}_{p1}{N}_{p2}f/(2\pi {\Sigma }_{\mathrm{x}}{\Sigma }_{\mathrm{y}}),$$

$$L_{\mathrm{specific}}=L_{\mathrm{TPX}}/(N_{\mathrm{b}}{N}_{p1}{N}_{p2})=f/(2\pi {\Sigma }_{\mathrm{x}}{\Sigma }_{\mathrm{y}}).$$

$$n_{\mathrm{f}}=\frac{2.724\text{SIUnitSymbolMicro}{\mathrm{b}}^{-1}{\mathrm{s}}^{-1}}{2386\mathrm{hit}{\mathrm{s}}^{-1}}=1.141\cdot 10^{-3}\text{SIUnitSymbolMicro}{\mathrm{b}}^{-1}/hit.$$

$$\mu =L\cdot {\sigma }_{\mathrm{inel}}/(N_{\mathrm{b}}\cdot f),$$

$$-dN/dt={\lambda }_{\mathrm{bb}}{N}^2/N_0+{\lambda }_{\mathrm{g}}N,$$

$$N(t)=\frac{N_0{\mathrm{e}}^{-{\lambda }_{\mathrm{g}}t}}{1+\frac{{\lambda }_{\mathrm{bb}}}{{\lambda }_{\mathrm{g}}}(1-{\mathrm{e}}^{-{\lambda }_{\mathrm{g}}t})},$$

$$N(t)=N_0{\mathrm{e}}^{-{\lambda }_{\mathrm{g}}t}\mathrm{for}{\lambda }_{\mathrm{bb}}\ll {\lambda }_{\mathrm{g}}\mathrm{and},$$

$$N(t)=\frac{N_0}{1+{\lambda }_{\mathrm{bb}}t}\mathrm{for}{\lambda }_{\mathrm{g}}\ll {\lambda }_{\mathrm{bb}}.$$

$$t_{\mathrm{bb}}^{\mathrm{inel}}=N_{\mathrm{b}}{N}_0/(N_{\exp }{L}_0{\sigma }_{\mathrm{inel}}),$$

$${\lambda }_{\mathrm{bb}}^{\mathrm{inel}}=1/t_{\mathrm{bb}}^{\mathrm{inel}}=0.2815{\mathrm{day}}^{-1}.$$

$${\lambda }_{\mathrm{bb}}^1=\sqrt{10.51/24.8}\cdot 0.2815{\mathrm{day}}^{-1}=0.1833{\mathrm{day}}^{-1}.$$

$${\lambda }_{\mathrm{g}}^{\mathrm{avg}}=(0.04\pm 0.03)\mathrm{da}{\mathrm{y}}^{-1},$$

Full text

I Introduction

A TPX detector network [1] of sixteen devices was installed in the ATLAS cavern at CERN. Each TPX device consists of two stacked hybrid silicon pixel sensors. The silicon sensors have a matrix of $256\times 256$ pixels of $55\,\micron$ pitch, and thickness of $300\,\micron$ (further indicated as layer-1) and $500\,\micron$ (layer-2) [2]. The readout chips connected to these sensors have the original Timepix design [3, 4]. The installation of the TPX devices took place during the LHC shutdown transition from Run-1 to Run-2 in 2013/2014. These double-layer TPX devices replaced the previously operational network that employed single-layer Medipix (MPX) assemblies [5, 6].

These devices measure the primary and secondary particle fluxes resulting from 13 TeV proton-proton collisions. The data were taken in 2015 during the first year of LHC Run-2 operation. Precision luminosity measurements are of particular importance for many physics analyses in high-energy physics.

The use of the TPX network for luminosity measurements has several advantages compared to the previous luminosity measurements [7] at LHC during Run-1 that used MPX devices. Regarding the luminosity monitoring, the two-layer hodoscope structure of the TPX devices doubles the measurement statistics and allows one to determine the precision and long-term time stability of individual TPX devices. The dead-time caused by the readout was reduced from about 6 s to 0.12 s allowing a much higher data acquisition rate. Also, the TPX devices are operated in three different modes [3, 4]: hit-counting, time-over-threshold (energy deposits and cluster-counting), and time-of-arrival (cluster-counting). Furthermore, the proton-proton collision energy increased from 8 TeV at LHC during Run-1 to 13 TeV at Run-2 which opened a new energy frontier for luminosity measurements in colliders.

The TPX network is self-sufficient for luminosity monitoring. It collects data independently of the ATLAS data-recording chain, and provides independent measurements of the bunch-integrated LHC luminosity. In particular, van der Meer (vdM) scans [8] can be used for absolute luminosity calibration.

The detection of charged particles in the TPX devices is based on the ionization energy deposited by particles passing through the silicon sensor. The signals are processed and digitized during an adjustable exposure time (frame acquisition time) for each pixel. Neutral particles, namely neutrons, however, need to be converted to charged particles before they can be detected. Therefore, a part of each silicon sensor is covered by 6LiF and polyethylene converters [9, 2].

Thirteen out of the sixteen installed devices have been used for the luminosity analysis. Two devices were found to be inoperational after the closing of the ATLAS detector and one device was intentionally located far away from the interaction point and therefore it was unusable for luminosity measurements. Table I lists the locations of the devices TPX02 and TPX12 used in this analysis, and their numbers of registered passing particles (clusters). It is noted that the number of clusters for the $500\,\micron$ sensor is about 20% larger compared to the $300\,\micron$ sensor. This percentage is lower than might be expected from the 40% larger sensitive volume, because the extended clusters induced by a single particle count only ‘one’ in the thin as well as in the thick sensor. The number of photon conversions and fast neutron interactions, however, will increase with the sensitive volume. The analysis described in this article is focused on the precision luminosity determination with the devices TPX02 and TPX12, which are operated with 1 s exposure time and analysed in hit-counting mode. As their positions are very similar in $R$ and $Z$ coordinates on opposite sides of the proton-proton interaction point, their count rates are very similar. During 2015 LHC proton-proton collisions typical luminosities were $\rm{\cal L}=$ 3-5$\rm\cdot 10^{33}\,cm^{-2}s^{-1}=$ 3000-5000$\rm\,\mathrm{\SIUnitSymbolMicro}b^{-1}s^{-1}$ and the TPX count rate was a few $10^{6}$ hits/s per frame.

Figure 1 shows an example of the luminosity from hit-counting measured with TPX02 layer-2 for LHC fill 4449, taken on 2-3 October 2015. All times are in GMT.

This article is structured as follows. First, the concept of LHC luminosity monitoring from TPX hit-counting is introduced in Section II. The vdM scan analysis based on hit-counting (Section III) is summarized for an absolute luminosity calibration. In Section IV the LHC luminosity curve is determined by using the concept of averaged interactions per bunch-crossing, together with the TPX measurement precision. Section V describes the luminosity precision by evaluation of the difference between two layers of the same TPX device. The long-term luminosity precision is given in Section VI from the comparison of layer-1 and layer-2 luminosity of the same TPX device. The long-term luminosity precision from different TPX devices is given in Section VII. Finally, conclusions are given in Section VIII.

II LHC Luminosity from TPX Hit-Counting

The data from the TPX02 and TPX12 devices were used in hit-counting mode, both having similar count rates as specified in Table I. The devices measure the luminosity independently and their measurements are cross-checked with each other. A constant exposure time of 1 s was used for the entire 2015 data-taking.

A small number of pixels becoming weak or noisy (e.g. due to radiation damage) could have a significant effect on the luminosity measurement. Therefore, pixels with a count rate that is at least $3\sigma$ away from the mean, are excluded for each sensor region (uncovered and with converters). This requirement identifies about 7-13% of the pixels on layer-1 and layer-2 both for TPX02 and TPX12 per LHC fill, including 5-10% of pixels at the boundaries of the sensor regions and edges of the sensor matrix. Then, the logical OR of identified pixels per LHC fill was taken for all 2015 LHC fills to remove 19-22% of the total number of pixels of the sensors. The effect of the pixel removal on the analysis was also studied with $2\sigma$ and $5\sigma$ criteria, with the result that the analysis outcome regarding the LHC luminosity curve, measurement precision and long-term stability remained unchanged.

The hit rate for the four TPX sensors is normalized to units of luminosity by multiplying with a scaling factor as given in Section III.

The induced radioactivity of material in the ATLAS cavern has no significant effect on the luminosity determination as determined by a dedicated study.

The ATLAS and CMS collaborations have elaborate systems of luminosity measurements, described in [10], [11] (ATLAS) and [12], [13] (CMS). A comparative study of their results and the TPX luminosity monitoring is beyond the scope of this article.

In addition to the hit-counting, luminosity can be measured with the two other modes of TPX operation based on cluster-counting and summed energy deposits. These luminosity measurements will be addressed in a separate publication.

The relation between the number of hits and clusters (particles) is investigated in order to determine the statistical uncertainty of the luminosity measurement from hit-counting. The average ratio of hits per cluster is approximately $R=N_{\rm hit}/N_{\rm cl}=10$, which was obtained with TPX data from a low-intensity LHC fill for which the clusters on the sensors were well separated. This factor is used for the statistical uncertainty determination in the following sections assuming that one cluster corresponds to one independent particle passing the device [7].

III Van der Meer Scans

Van der Meer (vdM) scans are used for absolute luminosity calibration at the LHC. This scan technique was pioneered by Simon van der Meer at CERN in the 1960s [8] to determine the luminosity calibration in a simple way. It involves scanning the LHC beams through one another to determine their sizes in terms of the horizontal and vertical widths of the beams at the point of collision. These width measurements are then combined with information of the number of circulating protons, allowing the determination of an absolute luminosity scale. The vdM scan analysis is based on the data taken on 24-25 August 2015 using TPX02 and TPX12 layer-1 and layer-2.

The LHC beam separation dependence of the measured TPX luminosity is well represented by the sum of a single Gaussian and a constant (Fig. 2). The absolute luminosity normalization is derived from the combination of the hit rate, the horizontal and vertical convoluted widths and the average bunch currents.

The measurement uncertainty of the TPX devices can be determined with respect to the expected statistical uncertainty. For this study, the pull distributions, as defined by (data-fit)/$\sigma_{\rm data}$, were determined, where $\sigma_{\rm data}=\sqrt{R}\cdot\sigma_{\rm stat}^{\rm hit}$, $\sigma_{\rm stat}^{\rm hit}={\rm data}/\sqrt{N_{\rm hit}}$ and $R=10$. Figure 3 shows the pull distribution for the first horizontal vdM scan in August 2015, as seen by TPX02 layer-2. The sigma of the pull distribution averaged over TPX02 and TPX12, layer-1 and layer-2 for both horizontal and vertical scans is $2.0\pm 0.6$, which indicates that additional uncertainties are present beyond the determined statistical uncertainties, or correlations in the statistical evaluation have a significant effect as discussed in Ref. [7]. Furthermore, transverse proton-bunch profiles are not expected to be perfectly Gaussian; and even if they were, a scan curve summed over Gaussian bunches of different widths would not be strictly Gaussian. Therefore, non-Gaussian contributions to the vdM-scan curves may contribute, at some level, to the widening of the pull distribution.

For TPX02 layer-2, the widths of the beam sizes (horizontal and vertical nominal beam separations) and their statistical uncertainties are $\Sigma_{\rm x}=(130.2\pm 0.5)\,\micron$ and $\Sigma_{\rm y}=(118.6\pm 0.5)\,\micron$, respectively.

The luminosity can be calculated as:

[TABLE]

where $N_{\rm b}$ is the number of bunch crossings producing collisions per machine revolution, $N_{\rm p1}$ and $N_{\rm p2}$ are the average bunch populations (number of protons) in beam 1 and beam 2, respectively, $f$ is the machine revolution frequency (11245.5 Hz), and $\Sigma_{\rm x}$ and $\Sigma_{\rm y}$ are the convoluted horizontal and vertical beam sizes. The LHC parameters for fill 4266 are [14]:

• •

number of bunches: $N_{\rm b}=30$;

• •

average number of protons (in units $10^{11}$) per bunch in beam 1 and in beam 2: $N_{\rm p1}=26.5/30=0.883$ and $N_{\rm p2}=26.6/30=0.887$, respectively.

Thus, the resulting luminosity is $L_{\rm TPX}=2.724\,{\rm\mathrm{\SIUnitSymbolMicro}b^{-1}s^{-1}}$.

The specific luminosity is defined as:

[TABLE]

Table II summarizes the scan results for the first pair of vdM scans (horizontal and vertical) registered with TPX02 and TPX12 for both their layers.

For TPX02 layer 2, the fits of horizontal and vertical scans provide $(2357\pm 14)$ and $(2415\pm 16)$ hits/s, respectively, at the peak above the background. The average number is ($2386\pm 11$) hits/s. Thus, the normalization factor $n_{\rm f}$ between the TPX02 layer-2 hit rate and the instantaneous LHC luminosity is

[TABLE]

The normalization factors for the other devices were calculated using the same procedure, and the results are summarized in Table III.

As already noted for the previous LHC Run-1 vdM scan analysis using MPX devices [7], the normalization factor for the absolute luminosity is only approximate since the TPX exposure time is much longer than the bunch spacing. Therefore, the bunch-integrated luminosity averages over the different bunch profiles. In order to estimate the resulting uncertainty a simulation with 29 overlapping Gaussian distributions was performed [7], which led to an estimate of the resulting uncertainty on the normalization factor (from this source only) of about 1%.

Although further uncertainties could arise from non-Gaussian shapes, this study shows that the Gaussian approximation of the sum of Gaussians is quite robust with the TPX system and the luminosity approximation by bunch integration is a sensible approach. No attempt was made for a precise determination of the total uncertainty which would require a dedicated study [10, 11].

IV LHC Luminosity Curve and TPX Precision

The TPX network has the capability to study the LHC luminosity curve with precision. Six LHC fills of proton-proton collisions were investigated in detail. As an example, details are given for the LHC fill 4449, taken on 3 October 2015.

First, the TPX luminosity is calculated using the normalization factors from Table III and it is then converted to an average number of inelastic interactions per bunch crossing by

[TABLE]

where $N_{\rm b}=1453$ colliding bunches, $f=11245.5$ Hz and the inelastic cross-section $\sigma_{\rm inel}=80$ mb [15]. Accelerator simulations [16] have shown that under routine physics conditions ($\beta^{*}<1$ m), elastic proton-proton scattering contributes negligibly to the particle-loss rate. This is because the typical scattering angle is so small compared to the natural angular divergence of the beam that the protons remain with the dynamic aperture of the ring.

IV-A Fitted LHC Luminosity Curve

In a simple approximation, the loss rate of protons $N$ in the colliding beam is governed by:

[TABLE]

where $N_{0}$ is the initial number of protons, and $\lambda_{\rm bb}$ and $\lambda_{\rm g}$ are constants related to beam-beam (burning-off the proton bunches) and single bunch (e.g. beam-gas) interactions, respectively. This equation has a known solution, which is used as fit function:

[TABLE]

with two well-known border cases:

[TABLE]

[TABLE]

Next, the expected mean lifetime of inelastic beam-beam interactions is calculated from the LHC beam parameters, as given in Table IV, and used as a fixed parameter to determine $\lambda_{\rm g}$. A simultaneous fit of $\lambda_{\rm bb}$ and $\lambda_{\rm g}$ did not converge. As already noted in the previous LHC Run-1 analysis [7], $\lambda_{\rm bb}$ and $\lambda_{\rm g}$ are strongly anti-correlated. Compared to the data-taking at Run-1, the Run-2 LHC luminosity curve is flatter, making the fit less sensitive to the fit parameters.

The mean lifetime from inelastic beam-beam interactions is given by [17]:

[TABLE]

where $N_{0}$ is the initial number of protons per bunch ($N_{\rm b}N_{0}=1.6\cdot 10^{14}$ protons [14]). For LHC fill 4440 the initial luminosity is $L_{0}=3360\,{\rm\mathrm{\SIUnitSymbolMicro}b^{-1}s^{-1}}$ [14], the number of experiments is $N_{\rm exp}=2$ (ATLAS [18] and CMS [19]). For the lifetime $t_{\rm bb}^{\rm inel}=3.07\cdot 10^{5}\,{\rm s}$ is obtained, thus

[TABLE]

The value $\lambda_{\rm bb}^{\rm inel}$ depends on the initial luminosity and the initial number of protons, thus on the starting value of $\mu_{1}=10.43$ for the first fit. Since $L\propto N^{2}$, one can write $\lambda_{\rm bb}^{\rm inel}\propto\sqrt{L_{0}}\propto\sqrt{\mu_{0}}$. Thus for the lower initial luminosity in the fit, a longer lifetime is expected from beam-beam interactions and therefore a smaller

[TABLE]

The frequent LHC small-amplitude beam-separation scans for optimisation of the luminosity made it necessary to adapt the fit function. Therefore, the data during the scans is removed and the LHC luminosity curve is fitted with a function having the values of $\lambda_{\rm bb}$ reduced by $\sqrt{\mu_{i}/\mu_{0}}$ for each time period between the scans. The $\mu_{i}$ values are calculated for the starting values for each region between the scans, and $\mu_{0}$ is the value for the start of the LHC fill.

Six long LHC fills with large luminosity are investigated as given in Table IV. Figure 4 shows the fit of TPX02 layer-2 data for two of these LHC fills.

The fit results indicate that the LHC luminosity reduction is predominantly reduced by beam-beam interactions since a larger value of $\lambda_{\rm bb}$ corresponds to a shorter lifetime. In order to determine the $\lambda_{\rm g}$ and its uncertainty, the fit is performed for the six LHC fills and the four sensors separately, resulting in a large single-bunch lifetime $1/\lambda_{\rm g}$:

[TABLE]

where the average is taken over 24 measurements, and the uncertainty is given as the square root of the variance.

IV-B Precision

In order to investigate the precision of the TPX luminosity measurements, the difference between the fit and the data is studied as a function of time. As an example, the data is analysed from the last continuous LHC curve taken on 3 October 2015 for about 4.5 hours starting 0:45.

A precision of 0.35% has been obtained with TPX02 layer-2 and similarly with other TPX sensors, as listed in Table V. The given uncertainties result from statistical and systematic uncertainties by the TPX measurements convoluted with uncertainties arising from fluctuations in the proton-proton collision rates.

It is noted that the TPX measurement precision is statistically limited by the number of hits per frame. Therefore, 10, 20, 30 and 40 frames are grouped together. Consequently, the statistical precision significantly increases. Figure 5 shows the residuals for groups of 30 frames. The four distributions show no particular uniform structure and therefore no LHC luminosity variation at the 0.1% level. Table V lists the obtained precisions for TPX02 and TPX12 layer-1 and layer-2. The fit results using the data of the four sensors are consistent.

In order to increase the statistical significance of each TPX device the luminosity measured by layer-1 and layer-2 is averaged, and presented in Fig. 6 for 10 frames combined. It shows the relative difference between data and fitted average number of interactions per bunch crossing as a function of time, seen by TPX02. The corresponding Gaussian fit is shown in Fig. 7. Figure 8 shows the pull distribution assuming statistical uncertainties only. Table VI lists the obtained uncertainties and the corresponding pull values for 1  frame, as well as for 10, 20, 30 and 40 frames combined. The resulting relative short-term luminosity measurement precision is 0.1% for 10 s time intervals. As a consistency check of the obtained precision, the data from 7 October 2015 (LHC fill 4467) from 5:30 to 10:00 was analysed in the same way, reproducing the results on the precision. Thus, the resulting luminosity precision of the TPX system is higher compared to the previous MPX luminosity measurement precision (0.3% for 60 s time interval) [7] for data taken at LHC during Run-1 operation.

V Short-term Precision of Individual TPX Devices

In order to determine the short-term precision of individual TPX devices for luminosity measurements, the relative difference in luminosity measured by layer-1 and layer-2 of the same TPX device is studied as a function of time. The statistical precision is increased by grouping 10, 20, 30 and 40 frames. Figure 9 shows the fit results for 10 frames combined, while Fig. 10 shows the corresponding precision as the width of the Gaussian fit. The resulting pull distribution is shown in Fig. 11. For single frames the width of the pull distribution is about unity, indicating that the statistical uncertainty is dominant. The width of the pull distribution is increasing as more frames are combined in order to decrease the statistical uncertainty, and the pull values indicate that in addition systematic uncertainties are present.

Table VII lists the obtained precisions for TPX02 and TPX12. As the difference of two measurements (from layer-1 and layer-2) is calculated and the statistical significance of each measurement is about the same, the uncertainty of each measurement is about $\sqrt{2}$ of overall uncertainty, thus leading to a measurement precision for each TPX device of approximately 0.1% for 10 s time intervals.

VI Long-term Stability of Individual TPX Devices

The long-term time stability of the luminosity monitoring is determined for individual TPX devices by comparing the luminosity measured by the two separate sensitive layers of TPX02 and TPX12. For this analysis the frames are grouped corresponding to time periods of the 2015 LHC fills with instantaneous luminosity above $\rm 1000\,\mathrm{\SIUnitSymbolMicro}b^{-1}s^{-1}$. A linear fit is applied to the (TPX layer-1)/(TPX layer-2) luminosity ratio versus time for the August to November 2015 data-taking period, as given in Fig. 12. The slope of the linear fit is taken as a measure of time stability. The uncertainty is obtained from the fit. As the statistical uncertainty of the ratio measurements for the grouped frames is much less than the systematic uncertainties, each ratio is given equal weight in the fit. The obtained slope values and their uncertainties for TPX02 and TPX12 are summarized in Table VIII. The slope shows that the luminosity ratio measured by layer-1 and layer-2 slightly decreases with time. This could indicate a small relative change in sensitivity with time between the thinner and thicker sensors. The time-stability of the luminosity measurements between individual layers is about 0.5% per 100 days.

In addition, a study of the internal calibration transfer was performed. As described in Section III the normalizations of the TPX sensors were performed with a vdM scan using LHC fill 4266 (24-25 August 2015) with a peak luminosity of $\rm 2.7\,\mathrm{\SIUnitSymbolMicro}b^{-1}s^{-1}$. In the same LHC fill, the luminosity was kept constant after the vertical and horizontal scans for about 5 hours. Using the data of this time period the (TPX layer-1)/(TPX layer-2) luminosity ratio was determined with TPX02 and TPX12, and found to be in agreement with the ratios at high luminosity within 0.5% uncertainty.

VII Long-term Stability of Different TPX Devices

The long-term time stability of the luminosity monitoring is determined for two TPX devices located about 7 m apart at opposite sides of the primary proton-proton interaction point. First, the frames are grouped into luminosity blocks of about one minute. This grouping of frames is necessary for a comparative study as the start times of the frames are not synchronized between the TPX devices. Then, the luminosity blocks are grouped into time periods corresponding to the LHC fills with instantaneous luminosity above $\rm 1000\,\mathrm{\SIUnitSymbolMicro}b^{-1}s^{-1}$. The luminosity ratio measured by TPX02 and TPX12 is calculated and a linear fit is applied for the August to November 2015 data-taking period, as given in Fig. 13. The slope of the linear fit is taken as a measure of time stability. The obtained slope value and its uncertainty is summarized in Table IX. The uncertainty is obtained from the fit. As the statistical uncertainty of the ratio measurements for the grouped time intervals is much less than the systematic uncertainties, each ratio is given equal weight in the fit. The fluctuations are much larger than the slope of 0.02% per 100 days, therefore, the largest fluctuations are used as an estimate of the long-term time stability. These fluctuation could either result from the TPX operation, or from small variations in the complex LHC radiation field depending on small changes in the colliding beam optics. Thus, conservatively, the study of two different TPX devices indicates an internal time stability of the luminosity measurement below 0.5% per 100 days.

VIII Conclusions

The network of TPX devices installed in the ATLAS detector cavern has successfully taken data at LHC during Run-2 with 13 TeV proton-proton collisions. An approximate absolute luminosity calibration was determined from a vdM scan in 2015. The TPX network measured the LHC luminosity curve with precision indicating that the luminosity reduction from single-bunch interactions was much less than from beam-beam interactions. The relative short-term precision of the TPX luminosity measurements was determined to be 0.1% for 10 s time intervals, and the internal long-term stability of the TPX system for luminosity measurements was below 0.5% for the 2015 data-taking period.

Acknowledgment

The authors would like to thank warmly the ATLAS Luminosity Group for useful discussions and interactions, and the Medipix Collaboration for providing the TPX assemblies. The project is supported by the Ministry of Education, Youth and Sports of the Czech Republic under projects number LG 15052 and LM 2015058, and the Natural Sciences and Engineering Research Council of Canada (NSERC). Calibration measurements were performed at the Prague Van-de-Graaff accelerator funded by the Ministry of Education, Youth and Sports of the Czech Republic under project number LM 2015077.