Study of energy response and resolution of the ATLAS Tile Calorimeter to hadrons of energies from 16 to 30 GeV

Three spare modules of the ATLAS Tile Calorimeter were exposed to test beams from the Super Proton Synchrotron accelerator at CERN in 2017. The measurements of the energy response and resolution of the detector to positive pions and kaons and protons with energy in the range 16 to 30 GeV are reported. The results have uncertainties of few percent. They were compared to the predictions of the Geant4-based simulation program used in ATLAS to estimate the response of the detector to proton-proton events at Large Hadron Collider. The determinations obtained using experimental and simulated data agree within the uncertainties.


Introduction
Three spare modules of the Tile Calorimeter (TileCal) of the ATLAS experiment [1], two long-barrels and one extended-barrel, were exposed to muons, electrons, pions, kaons and protons with different energies and incident angles at test beams (TBs) in 2017 [2]. The role of the hadron calorimetry in ATLAS is to measure the energy and the angle of isolated hadrons and jets. To achieve good performance, the study of the sub-detector response to isolated hadrons is important. In this paper, the measurements of the calorimeter response and resolution to positive pions and kaons and protons, with energies in the range [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] GeV are presented. The results are compared with the ones obtained analyzing simulated data produced using the ATLAS Geant4 toolkit [3], [4] and [5]. The experimental setup including the beam line counters and the detector is described in Section 2. The data sets, the event selections and the reconstruction of the particle energies in the case of experimental and simulated data are presented in Sections 3 and 4, respectively. The determinations of the calorimeter responses and resolutions are discussed in Section 5. The results are compared with hadronic cascade model predictions in Section 6. The conclusions are stated in Section 7.

The beam line
The measurements discussed in this paper were performed using tertiary particle beams at the H8 line in the North Area of CERN [2]. Secondary beams are produced by targeting 400 GeV protons, from the Super a Corresponding author e-mail: tigran.mkrtchyan@cern.ch b Corresponding author e-mail: claudio.santoni@cern.ch   Table (on which was placed the TB calorimeter) setup are shown.
Proton Synchrotron (SPS) accelerator, on a 100 mm thick T4 target made of beryllium (primary target). Using Secondary Targets located at about 130 m downstream of the T4 target, tertiary beams can be produced. A large spectrometer constructed of four Main Bend North Area dipole magnets is used for the momentum definition. Beam particles can have energies from 10 to 350 GeV. Beam intensity decreases dramatically at the low energies. To have mixed hadron enriched tertiary beams, the Secondary Target is made of copper and has a thickness of 300 mm. Additionally, a lead absorber (6 mm) is moved into the beam about 270 m downstream of the target. It absorbs the electrons, while the hadrons mostly pass through it. For electron enriched tertiary beams, the Secondary Target is made by aluminum and has a thickness of 400 mm. It is immediately followed by 6 mm of lead. The lead absorber further downstream is moved out of the beam trajectory.
The layout of the beam line detectors is shown in Figure 1. The transverse beam profile was monitored by the wire chamber BC1 [6]. Two scintillating counters, S1 and S2 with an active surface of 5 × 5 cm 2 [7], were used in coincidence to trigger the data acquisition (Physics Trigger) and to provide the trigger timing. These two detectors were also used to reject beam particles interacting upstream of the detector. The Cherenkov counters Ch1, Ch2 and Ch3 allowed identification of beam particles. The counters Ch1 and Ch3 distinguish electrons and pions from kaons and protons. They were filled with CO 2 and He, respectively. The pressure values set for the different beam energies are reported in Table 1. The Cherenkov counter Ch2 was also filled with CO 2 . The higher pressure in Ch2 allows for separation of kaons from protons. More details can be found in Ref. [7].

The detector
The TB setup, shown in Figure 2, consists of three spare ATLAS modules [1] of TileCal, two long-barrels and  one extended-barrel, stacked on a scanning table (see Figure 1) that is capable of placing modules at different position and angle with respect to the incoming beam particles. An extended-barrel (long-barrel) consists of one (two) super-drawer(s). In the figure they are named M0A and M0C (module at the bottom), LBA 65 and LBC 65 (module in the middle) and EBC 65 (module at the top). Some of the super-drawers were equipped with different upgraded front-end electronics systems proposed for the ATLAS LHC Phase-II operations [8]. The super-drawers EBC 65 and M0 C were equipped with the electronics installed currently in AT-LAS [1]. As shown in Figure 3, the modules have a periodic structure of steel plates and scintillating tiles perpendicular to the z axis. Wavelength-shifting fibres transmit light produced in the tiles to the PMs [9]. In each module a three-dimensional cell structure is defined by grouping optical fibres connected to the same PM [10]. In general two PMs read-out a cell and the signals are summed up to provide the cell response. A structure of three cell layers parallel to the z axis is obtained. The cell layers A, BC and D in half long- Fig. 3: Mechanical structure of a TileCal module, showing the slots in the steel for scintillating tiles and the method of light collection by wavelength-shifting fibres to PMs. The holes for radioactive source tubes that traverse the module perpendicularly to the iron plates and scintillating tiles are also shown.
barrel and A, B and D in extended barrel are shown in Figure 4.
As in the ATLAS detector at LHC, the energy deposited in a cell of the TB detector, E raw c , was determined making use of the Optimal Fit method [11]. The linearity of the ADC's is determined using the Charge Injection System (CIS) [12]. The inter-calibration of the different calorimeter cells was obtained by equalizing the PM current induced by movable radioactive 137 Cs sources that cross every row of scintillating tiles near the edges (see Figure 3). Since the scintillating tile response depends on the impact point position of the particle in the tile and on the tile size, correction factors were applied for each layer of the calorimeter. Those values were determined from 1990's Test Beam data, which measured the response to muons impinging on the calorimeter with a direction parallel to the z axis (see Figure 2), and from the measurements obtained using a Sr source [12]. The scale of the reconstructed cell energy, C EM c = 1.05 pC⁄GeV, was obtained using electron beams incident at the centre of each cell with an angle of 20 • with respect to the cell surface normal. The estimated uncertainty is ∆C EM c = 2.4% [12]. The analysis of the muon and electron test beam data collected in the 2017 Test Beam [8] produced performance results that agree with the ones obtained using previous TBs [12] and with in-situ measurements in ATLAS [13].
To be consistent, the Optimal Fit method [11] was applied also to reconstruct the energy deposited in the cells in the case of simulated events. The scale of the cell energy measurements was obtained using the response to simulated electrons The energy deposited by the beam particles incident the detector, E raw , was determined as the sum of the energy measured in the calorimeter cells.

Analysis of experimental data
The results discussed in this paper were obtained exposing the TB calorimeter setup to enriched tertiary positive hadron beams with energy, E beam , equal to 16, 18, 20 and 30 GeV. As shown in Figure 2, the beams hit at the middle of the cell A3 of the super-drawer LBC65 with an azimuth angle φ = 0 and polar angle θ of about 76°, corresponding to a pseudo-rapidity values η = 0.25 [1]. (see Figure 4). The angle from the calorimeter module normal is equal to 14 degrees. The numbers of events collected during the data taking period are reported in Table 2 (Physics Trigger). 3.1 Collimated single-particle events Collimated single-particle events were first selected using beam detectors upstream of the TB calorimeter setup. The selection criteria on the beam line scintillating counters signals, E S1 and E S2 , were established making use of the responses of S1 and S2 to muons. Muon events were recognized by requiring an energy deposited in the module LBC65 compatible with the one deposited by a minimum ionizing particle. The retained events satisfy the criteria: and where the quantities E m.p.

S2
(µ) are the most probable (m. p.) values of the S1 and S2 muon signal distributions respectively. The selection criteria, especially useful for electron studies, remove particles that initiated a shower upstream of the calorimeter, as well as multi-particle beam events. The number of events retained after the application of the criterion are reported in Table 2 (Selection 1.). Events with a beam trajectory far away from the beam axis were rejected because the beam particles might have scattered upstream and therefore be off-energy. The beam chamber BC1 allows a determination of the transverse beam impact point coordinates, x BC1 and y BC1 . Gaussian functions were fitted to the distributions of each data set to determine the peak values x peak BC1 and y peak BC1 respectively. The accepted events have the beam impact point coordinates inside the square surface of the trigger scintillating counters: and |y BC1 − y peak BC1 | < 2.5 cm.
The numbers of events retained after the application of this criterion are reported in Table 2 (Selection 2.).

Muon rejection
The second set of criteria allows identifying pure samples of hadrons. As already mentioned, at the considered beam energies, muons are minimum ionizing particles and deposit in the scintillating tiles energy much smaller than electrons and hadrons (see Figure 5). The muon rejection was obtained requiring a reconstructed energy in the detector (see Section 2.2) E raw E raw µ cut = 5 GeV. The selection criterion allows also a rejection of spurious trigger events. The retained events are reported in Table 2 (Selection 3.). The events were selected applying the selection criteria up to Selection 2. (see Table 2). The muons and spurious events were rejected in the analysis requiring E raw larger than E raw µ cut = 5 GeV, as shown in the histograms.

Electron identification
As shown in Figure 6, the signals measured in Cherenkov counters Ch1 and Ch3, S Ch1 and S Ch3 , respectively, allow a separation of pions and electrons (e/π) from kaons K and protons p (K/p). The selection criteria in ADC counts applied on the signals are reported in Table 3. The numbers of the identified events are reported in Table 2. As discussed in Section 3.3, the Ch2 measurements allow separating kaons and protons. The electron components in e/π samples were determined statistically exploiting the difference of electromagnetic and hadronic shower profiles in the calorimeter modules [12]. Two separators, C long and C tot , were used: Table 3: Selection criteria in S Ch1 , S Ch2 and S Ch3 signals applied to identify e/π, K and p event samples for the four particle beam energy data sets. The Cherenkov signals are measured in ADC counts  Fig. 6: Scatter plots of the signals measured in the Cherenkov counter Ch3, S Ch3 , as a function of the signals measured in the Cherenkov counter Ch1, S Ch1 , in ADC counts. The histograms were obtained analysing data with beam energies equal to 18 GeV (a) and 30 GeV (b). The events were selected applying selection criteria summarized in Table 2, up to Selection 3. The cut values used to select kaon and proton, K/p, (left/bottom) and electron and pion, e/π, (right/top) events are shown. Colors are used in the plots to show the cell contents.
1. The shower profile parameter C long represents the fraction of the beam energy, E beam , deposited in the layers A of the modules (see Figure 4) : where i = 1, 2 and 3 indicate the super-drawers M0 C, LBC65 and EBC65 respectively. The parameter j runs over 3 contiguous cells of the three layers A around the cell hit by the beam and E raw c stands for the energy measured in a cell (see Section 2.2).
where N cell = 24 stands for the total number of contiguous cells, around the hit cell, considered for the shower profile estimate and the exponent α = 0.6 was tuned using a Monte Carlo (MC) simulation program to achieve maximum electron pion separation [12].
Scatter plots, C long vs C tot , of e/π sample events obtained using beams of particles with E beam equal to 18 and 30 GeV are shown in Figure 7. They can be compared with the ones in Figure 8 obtained using simulated electrons and pions events with the same beam energies. In general the pions have small values of C long and C tot , while in the case of electrons, the parameters have larger values localized in narrower regions. Pion events with large C long and C tot values are due to showers with large electromagnetic component. The analysis is based on the fact that electron (pion) C tot distributions are well described by one (two) Gaussian function. As an example, Figure 9 (a) shows the experimental C tot distribution obtained using an enriched electron beam with E beam = 20 GeV and C long ≥C min long = 0.6. The fit was performed in the region C tot ≥ 1.125. Figures 9 (b) and 9 (c) demonstrate that also simulated electron C tot distributions at 18 and 30 GeV are well described by one Gaussian function. Pion C tot distributions are best described by two Gaussian functions. The distributions of e/π data events with C long < C min long = 0.6 and E beam equal to 18   Table 2 are determined from the areas limited by such functions. The statistical uncertainties are equal to the corresponding diagonal terms of the fit error matrices. The events were selected requiring C long < 0.6. Two Gaussian functions fits, obtained using the method of the least squares, are superimposed on the data (red dashed curve). Red dotted curves show the individual Gaussian contributions.

Pion, Kaon and Proton identification
The third set of selection criteria was specific to the type of hadronic particles being studied. For each E beam data set the number of pions reported in Table 2 was estimated by subtracting the number of electron events obtained using the method described in Section 3.2.2 from the number of events of the corresponding e/π sample. The Ch2 signal measurements allow a separation of kaons and protons in the K/p samples. The scatter plots of the Ch2 signals, S Ch2 , in ADC counts units vs the energy measured in the calorimeter, E raw , obtained by analyzing data produced by beams of particles with energies equal to 18 and 30 GeV, are shown in Figure 13. The S Ch2 selection values in ADC count units are reported in Table 3. The obtained numbers of kaons and protons are reported in Table 2.

Reconstruction of the energy deposited in the modules
As already discussed in Section 2.2, the energy E raw deposited by incident particles in the detector was obtained as the sum of the energy measured in the calorimeter cells. In this study only cells with | E raw c | > 2σ noise were considered in the sum. For each run, the cell electronics noise σ noise was determined using random events collected between beam bursts (Pedestal Triggers). Typical noise values are of the order of 30 MeV. No corrections for dead material, containment and noncompensation effects were applied.
Due to the electron contamination, as sketched in the Figure 14, the pion energy distributions n π (E raw ) were obtained using, bin per bin, the formula n π (E raw ) = n e/π (E raw ) − N e f e (E raw ).
where n e/π (E raw ) is the number of e/π events in the considered E raw bin, the electron distribution f e (E raw ) is normalized to 1 and the number of electrons, N e , was determined using the procedure described in Section 3.2.2. Simulated electron distributions were used in the analysis because experimental data are available only for electron beam energy equal to 20 GeV. A comparison between the distributions obtained analyzing simulated and experimental electrons with the same beam energy, direction and impact point is shown in Figure 15 (a) . Figures 16 to 19 show the E raw distributions obtained in the case of beams of pions, kaons and protons with energies equal to 16, 18, 20 and 30 GeV, respectively.  4 Analysis of simulated data The experimental results obtained using positive pions and kaons and protons beams, with energies in the range 16-30 GeV, were compared to the predictions of the Geant4-based ATLAS simulation program [3], [4] and [5]. The FTFP BERT ATL hadronic showering model [14] was used in the simulation. This is the model presently being used in the simulation of the ATLAS events collected during the LHC Run 1 and Run 2. The number of generated events for each experimental data point is reported in Table 4. The responses of the beam line detectors were not included in the simulation. The distributions of the transverse beam impact point coordinates in the detector were tuned to reproduce the ones measured using the BC1. The TB detector material and geometry were fully described (see Ref. [4]). The measured electronics noise in the different calorimeter cells and the effects of photo-statistics (70 photo electron per GeV) in the PM signals, are included in the MC simulation. The simulated pion events were selected applying the C long and C tot cuts used in the analysis of exper- The black histograms correspond to the expected distributions of the electrons contaminating the samples obtained using simulated events. The normalization procedure is described in the text.
imental data. The numbers of the retained events for analyses are reported in Table 4. The shower energy was reconstructed using the same procedure applied in the case of experimental data. The distributions of E raw obtained using simulated data are shown in Figures 16 to 19 for beam energies equal to 16, 18, 20 and 30 GeV respectively.

Determination of the energy response and resolution
The experimental and simulated E raw distributions of pion, kaon and proton data are described reasonably well around the peak values by a Gaussian function. As in Ref. [12], the µ and σ parameters of Gaussian functions fitting the distributions in a region ±2σ around the peak values were used to estimate the measurement responses E raw and resolutions σ raw . An iterative procedure has been applied in order to get stable values of the parameters. The method of the least squares has been used. The fit functions obtained analysing experimental data are superimposed to the corresponding distributions in Figures 16 to 19. The fit results obtained using experimental and simulated data are reported in  (c) show the distributions of the reconstructed energy E raw obtained analysing simulated data obtained using electron beams with E beam equal to 20 GeV and 30 GeV respectively. The blue dot distribution in (a) has been obtained using experimental data. In (a) and (c) are also shown the distributions obtained using "high energy events" (red dashed line) and "low energy events" (red dotted line) discussed in Section 5.1. The histograms (b) and (d) show the oscillation of the electron response due to the sampling fraction variations as obtained using simulated electrons with E beam equal to 20 GeV and 30 GeV respectively. The dashed curves in red correspond to the fit of Eq. 10 to the data. The horizontal black line corresponds to the electron mean energy, p 0 (Eq.(10)). Table 5. The statistical uncertainties correspond to the square root of the corresponding diagonal term of the fit error matrix.

Energy responses and resolutions normalized to incident beam energy
Energy response normalized to incident beam energy and energy resolution normalized to incident beam energy  (π) and (σ raw (π))), kaons ( E raw (K) and (σ raw (K)) and protons ( E raw (p) and (σ raw (p))) with different beam energy. The statistical uncertainties correspond to the fit parameter uncertainties.   Table 6. In the case of experimental results, the first uncertainty value corresponds to the statistical uncertainty. The systematic uncertainty, second value, was obtained combining in quadrature the contributions of the seven sources discussed in the following. In the case of simulated data only statistical uncertainties are reported. Seven sources of systematic uncertainties were considered in the study: 1. Systematic Uncertainty 1. affects only pion determinations. It corresponds to the statistical uncertainty on the determination of the number of electrons contaminating the e/π samples discussed in Section 3.2.2. 2. As discussed in the same section the electron contamination was determined studying the C tot distributions of the e/π sample events with C long ≥ C min long = 0.6. Results obtained with different values of C long were used for uncertainty estimations. Systematic Uncertainty 2. values reported in Table 7 correspond to half of the differences of the determinations of R E raw and R σ raw obtained using C min long = 0.5 and C min long = 0.7 respectively. Table 6: Measured energy response (resolution) normalized to incident beam energy obtained using pions (R E raw (π) and (R σ raw (π))), kaons (R E raw (K) and (R σ raw (K))) and protons (R E raw (p) and (R σ raw (p))) of different beam energy obtained analyzing experimental and simulated data. In the case of experimental data, statistical and systematic uncertainties are reported. The effects of the different sources of systematic sources discussed in the text were combined in quadrature. Only statistical uncertainties are reported in the case of simulated data.  3. Effects due to the missmodeling of the C tot distributions used to determine the number of electrons contaminating the e/π samples was estimated comparing the results obtained using three Gaussian functions fits (see Section 3.2.2) with the ones obtained using two Gaussian functions fits. The estimated percentage of electrons increases from a value of 11% at 16 GeV up to 28% at 30 GeV. Systematic Uncertainty 3. values, affecting only pion determinations, are reported in Table 7 for each of the four beam energy samples. It is equal to the differences of the values of R E raw and R σ raw obtained using the two fitting functions. 4. As discussed in Section 3.4 the experimental E raw distributions of pions were obtained using Eq. (7). In Figure 15 are shown electron distributions obtained in the case of simulated data with beam energies equal to 20 and 30 GeV. Due to the regularly spaced scintillating tiles (see Figure 3) and the compactness of electromagnetic showers, the electron response varies with the periodicity of sampling fraction and thus depends on the coordinate of the impact point of the beam particles along the front face of the calorimeter module (z). In Figures 15 (b  and (d) is shown that the variation is reasonably well described by a simple periodic function [12] E raw (z) = p 0 [1 + p 1 sin(2πz/p 2 ) + p 3 ] .
The parameter p 0 corresponds to the mean reconstructed energy. The relative amplitude of the oscillation is described by p 1 . The parameter p 2 corresponds to the periodic thickness as seen by the beam at a given z value and p 3 is a phase. The behavior is responsible of the two peak structure of the E raw distributions evident, in particular, in the case of E beam = 30 GeV simulated data in Figure 15 (c). The effects of the uncertainty on the distribution of the z coordinates of the electron impact point on the determinations of R E raw and R σ raw was estimated using the E raw distributions of the events with a z value corresponding to E raw > p 0 , "high energy events", and E raw < p 0 , "low energy events", respectively. The distributions are shown in Figures 15 (a)   distributions. This uncertainty affects only pion determinations. 5. The 30 GeV scatter plot S Ch1 vs. S Ch3 in Figure 6 shows two spots in the K/p region. Their origin is not clear. Systematic Uncertainty 5. values reported in Table 7 correspond to the differences of the values of R E raw and R σ raw obtained using the events with S Ch1 ≤ 400 [ADC counts] and S Ch1 ≤ 250 [ADC counts], respectively. Although the other three energy data points do not show the two spot structure, a systematic uncertainty was determined also for them using the described procedure with the same selection criterion values. 6. As it appears in Figure 13, proton S Ch2 distributions show large tails. Their origin is not understood. Systematic Uncertainty 6. values in Table 7, correspond to the differences of the values of R E raw and R σ raw obtained using for each of the four proton beam energies, the upper values of the S Ch2 signals of Table 3, and the ones obtained selecting the events with S Ch2 ≤ 2000 ADC counts at 16 GeV, 18 GeV and 20 GeV and 1000 ADC counts at 30 GeV respectively. The same effect could also be present in the case of kaons. Since they produce a signal in Ch2, the effect may not be visible. For this reason the systematic uncertainty obtained for protons is also applied in kaon determinations. 7. The effect of the uncertainty of the scale of the reconstructed cell energy ∆C EM c on the measurements was also investigated. An estimation of the uncertainty on the energy response can be obtained using the formula: where ∆C EM c is equal to 2.4% (see Section 2.2) and E raw c i is the average energy deposited in the cell i. E beam is known at few per mile and one obtains the values of ∆ R E raw reported in Table 7 for the twelve data points (Systematic Uncertainty 7.). No significant dependence of the values on the beam energies was found. The uncertainty on C EM c affects in a negligible way the determinations of R σ raw .
The effects of each of the seven considered sources of systematic uncertainties on the four energy determinations are correlated. The uncertainty in the energy response normalized to incident beam energy is dominated by the systematic effects due to cell response non uniformity (Systematic Uncertainty 7.). Table 6 were obtained by combining in quadrature the effects of the seven sources reported in Table 7. Eleven values of the twelve energy response normalized to incident beam energy determinations have a total uncertainty smaller than 1.4%. It is mainly defined by the uncertainty in the calibration of the energy response of the relatively small part of the calorimeter involved in the study. In the case of kaons with E beam = 16 GeV, due to the large statistical error, the uncertainty on the determination of R E raw , is equal to 2.4%. Nine of the twelve determinations of the energy resolution normalized to incident beam energy, R σ raw , have a total uncertainty smaller than 1.9%. The uncertainty values of the determinations of R σ raw obtained in the case of 16 GeV pion and kaon and 18 GeV kaon beams are equal to 3.1%, 20.3% and 10.4% respectively.

The systematic uncertainties in
The determinations of R E raw (R σ raw ) as a function of E beam (1⁄ E beam [GeV]) are reported in the histograms of Figure 20 (Figure 21) . In the case of experimental results, statistical and systematic uncertainties are combined in quadrature. In the case of simulated results only statistical uncertainty are shown. Table 7: Systematic uncertainties on the estimations of R E raw and R σ raw in percent. The pion measurements are affected by the uncertainty on the number of electrons contaminating the e/π samples (Systematic uncertainties 1., 2. and 3.), on the E raw shape of the contaminating electrons (Systematic uncertainty 4.). The kaon and proton measurements are affected by the uncertainty on the Ch1 (Systematic uncertainty 5.) and Ch2 (Systematic uncertainty 6.) selection criteria. The uncertainty on the determination of the cell energy response non-uniformity, Systematic uncertainty 7., affects the determinations obtained for the three particle beams. 16 18 Syst. Beam Part. [

Comparison between experimental and simulated results
A quantitative comparison between experimental and simulated results can be obtained using the quantities The results are reported in Table 8 where statistical and systematic uncertainties are shown separately. The statistical uncertainties include the experimental and simulated uncertainties combined in quadrature. The results are also shown in Figures 20 and 21 where statistical and systematic uncertainties are combined in quadrature.
The average of the absolute values of the difference of all the energy response (resolution) measurements obtained using experimental and simulated data was found to be 1.1% (3.4%). In the case of the response determinations and the resolution determinations of pi- ons and kaons, the differences are consistent within the uncertainties. The uncertainties of the proton resolution determinations are about one order of magnitude smaller.

Comparison between pion, kaon and proton energy responses and resolutions
The values of the ratios R σ raw (K) R σ raw (π) obtained using experimental and simulated data are reported in Table 9. The statistical (first value) and the systematic (second value) uncertainties are shown separately in the case of experimental results. The systematic uncertainty was obtained combining in quadrature the contribution of the seven sources of systematic uncertainties discussed in Section 5.1 . The uncertainty on the scale of the reconstructed cell energy, C EM c , affects in a correlated way the reconstruction of the energy deposited in the modules by pions, kaons and protons. It follow that its effects on the energy response ratio determinations are negligible. In the case of simulated data only statistical uncertainties are reported. The determinations are also shown as a function of E beam in Figure 22. In the case of results obtained analyzing experimental data, the error bars were obtained combining in quadrature statistical and systematic uncertainties. In the case of results obtained analyzing simulated data only statistical uncertainty are shown. Table 9: Values of the ratios (14)-(17) obtained using experimental and simulated data produced by particles with E beam equal to 16, 18, 20 and 30 GeV. In the case of experimental determinations statistical (first value) and correlated systematic uncertainties (second value) are reported separately. Only statistical errors affect the MC determinations. In the considered E beam range, the measured ratios of the kaon over pion energy responses is constant with a weighted average equal to 0.967 ± 0.002 (-0.014). In parenthesis are reported the differences with the determinations obtained using simulated data. The rations of the energy responses of protons and pions range between 0.908 ± 0.008 (+0.009) at E beam = 16 GeV to 0.941 ± 0.001 (+0.010) at E beam = 30 GeV. The values of the ratios of the energy resolution determinations are constants. The weighted averages values are R σ raw (K)⁄R σ raw (π) = 0.95 ± 0.01 (-0.011) and R σ raw (p)⁄R σ raw (π) = 0.888 ± 0.005 (+0.011).
The results allow an extension down to 16 GeV of previous determinations of the ratios of the energy responses and of the resolutions of protons and pions obtained by ATLAS Collaboration using beams with energy above 50 GeV [12]. The response to protons was also reported systematically lower than that of negative or positive pions in Ref. [15]. The measurements were performed in the momentum range from 3 to 300 GeV/c. In the same paper results concerning the response of charged kaons and anti-protons in the momentum range below 9 GeV/c are reported. 16 Fig. 22: (a) Ratios of the kaon and proton responses over the pion ones as a function of E beam . The blue dots (black empty circles) show the ratios R E raw (K)⁄R E raw (π) obtained using experimental (simulated) data. The blue full (black empty) squares show the ratios R E raw (p)⁄R E raw (π) obtained using experimental (simulated) data. (b) Ratios of the kaon and proton resolutions over the pion ones as a function of E beam . The blue dots (black empty circles) show the ratios R σ raw (K)/R σ raw (π) obtained using experimental (simulated) data. The blue full (black empty) squares show the ratios R σ raw (p)⁄R σ raw (π) obtained using experimental (simulated) data. In the case of experimental results uncertainties include statistical and systematic effects combined in quadrature. In the case of simulated results only statistical uncertainty are reported.

Ratio
6 Comaparison with hadronic cascade model predictions 6.1 Parametrization of the energy response normalized to incident beam energy as a function of the beam energy The calorimeter response for pions, kaons and protons can be described in terms of the calorimeter non-compensation and leading particle effects [16]. The hadron energy response normalized to incident beam energy as a function of the beam energy can be parametrized according to where F h represents the non-electromagnetic energy component of showers induced by incident hadrons of energy E beam and e/h is the ratio between the responses to the purely EM and hadronic components of showers. The measurements allow a determination of the ratios of the non-electromagnetic energy component of showers induced by incident pions (F h (π)), kaons (F h (K)) and protons (F h (p)) for the same value of E beam . Using Eq. (18) one obtains and The determinations obtained using experimental and simulated data are reported in Table 10. The statistical (first value) and the systematic (second value) uncertainties are shown separately in the case of experimental results. The systematic uncertainties were obtained combining in quadrature the effects of the seven sources discussed in Section 5.1. In the case of simulated data, only statistical uncertainties are reported. Data show constant ratios F h (K)/F h (π). The weighted average numerical value is 1.13 ± 0.01 (1.072 ± 0.001). The values in parenthesis were obtained analysing simulated data. The ratio F h (p)/F h (π) decreases from 1.351 ± 0.04 (1.361 ± 0.003) at E beam 16 GeV to 1.24 ± 0.01 (1.281 ± 0.003) at E beam 30 GeV.
The ratio F h (p)/F h (π), as obtained in Refs. [17] and [18] from the copper/quartz-fiber calorimeter data [19], varies from 1.22 at 200 GeV to 1.15 at 370 GeV. In Ref. [20], a constant value of F h (p)/F h (π) in the range between 1.15 and 1.20 is predicted.
The determinations of F h (K)/F h (π) and F h (p)/F h (π) as a function of E beam are also reported in the histograms of Figure 23   In Groom's parametrization, [20], [17] and [18], one has where the quantity E 0 is the energy at which multiple pion production becomes significant and the parameter m describes the relation between the average multiplicity of secondary particles produced in the collision and the fraction of energy going into π 0 's in one collision. One obtains and Fits of Eq. (23) to the histograms of Figure 23 (a) and of Eq. (24) to the histograms of Figure 23 (b) allow a determination of  Fig. 23: (a) F h (K)/F h (π) as a function of E beam obtained using experimental, blue dots, and simulated data, black empty circles. (b) F h (p)/F h (π) as a function of E beam obtained using experimental, blue dots, and simulated data, black empty circles. In the case of experimental results uncertainties include statistical and systematic effects combined in quadrature. In the case of simulated results only statistical uncertainty are shown. The dashed (dotted) red curves are fits of the functions (23) and (24) to the experimental (simulated) data points. In case of experimental determinations the dashed red strips display the correlated systematic uncertainties. and respectively. The fit curves to the experimental and simulated data are show in the figure. The strips display the correlated systematic uncertainties ∆ syst. F h (K)/F h (π) (Figure 23 (a)) and ∆ syst. F h (p)/F h (π) (Fig. 23 (b)). They are defined by the curves obtained fitting Eq. (23) (Eq. (24)) to the points ). All the fits were performed using as uncertainties the statistical uncertainties of the determinations. The values of the parameters obtained in the fits are reported in Table 11. The first uncertainty value is the statistical uncertainty. It corresponds to the square root of the corresponding diagonal term of the fit error matrix. The systematic uncertainty (second uncertainty value) is equal to half of the differences of the determinations obtained fitting Eq. (23) to the points F h (K)/F h (π) ± ∆ syst. F h (K)/F h (π) and Eq. (24) to the points F h (p)/F h (π) ± ∆ syst. F h (p)/F h (π). In the table, the χ 2 probability values of the fits performed to the central points are reported. In the case of the fits to simulated data the probabilities are very small. Table 11: Values of the parameters B K/π (Eq. (25)) and C K/π (Eq. (26)) obtained fitting Eq. (23) to the experimental and simulated values of F h (K)/F h (π) as a function of E beam shown Figure 23 (a). Values of the parameters B p/π (Eq. (27)) and C p/π (Eq. (28)) obtained fitting Eq. (24) to the experimental and simulated values of F h (p)/F h (π) as a function of E beam shown Figure 23 ( The values of B K/π and C K/π obtained using experimental and simulated data agree within two sigmas. The values of B p/π and C p/π obtained using experimental and simulated data differ significantly. Fits of Eq. (22) to the determinations of R E raw as a function of E beam (see Figure 20) allow a determination [17] of m and The fit curves to the experimental and simulated determinations are reported in the figure. The strips display correlated systematic uncertainties ∆R E raw syst. . They are bounded by the curves obtained fitting Eq. (22) to the points R E raw ± ∆R E raw syst. . All the fits were performed using as uncertainties the statistical uncertainties of the determinations. The obtained values of A and m are reported in Table 12. The first uncertainty value is the statistical uncertainty. It corresponds to the square root of the diagonal term of the error matrix. The systematic uncertainty (second uncertainty value) is equal to half of the differences of the determinations obtained fitting Eq. (22) to the points R E raw ± ∆R E raw syst. . In the table the χ 2 probability values of the fits performed to the central points are reported. In the case of the fits to kaon and proton simulated data the probabilities are very small. The values of m obtained using pions, kaons and protons data without making any assumption on the values of e/h and E 0 are: 0.919 ± 0.005 (0.826 ± 0.005), 0.97 ± 0.02 (0.843 ± 0.002) and 0.789 ± 0.003 (0.734 ± 0.002) respectively. The values in parenthesis were obtained using simulated data. According to Ref. [17] values of m around 0.87 are expected. The determinations can be compared with previous pion measurements summarized in [17].
To compare the results discussed in this paper with the ones obtained previously using pions beams with energy in the range 10-350 GeV and incident in the Tile-Cal modules at η = 0.35 [12], Eq. (22) was fitted to the pion determinations fixing E 0 = 1 GeV. The obtained values e/h = 1.3535 ± 0.0304 and m = 0.9187 ± 0.0047 agree with the previous determination 1.33 ± 0.02 and 0.85 ± 0.03 respectively [12]. The uncertainties include statistical and systematic uncertainties combined in quadrature.

Parametrization of the energy resolution as a function of the beam energy
The resolution of the energy measurements as a function of the beam energy E beam can be parametrized according to where the first term describes the fluctuations on the number of particle produced in the showers, the second term describes the non-uniformity of the cell response and the symbol ⊕ indicates the sum in quadrature. In the considered beam energy range the noise contribution is negligible (see Section 3.4).
The curves in Figure 21 were obtained fitting Eq. (30) to the experimental and simulated determinations of R σ raw as a function of 1/ E beam [GeV]. The strips in the figure display correlated systematic uncertainties ∆R σ raw syst. . They are defined by the curves obtained fitting Eq. (30) to the points R σ raw ± ∆R σ raw syst. . All the fits were performed using as uncertainties the statistical uncertainties of the determinations. The resulting values of a and b are reported in Table 13. The statistical uncertainty (first uncertainty value) is equal to the square root of the corresponding diagonal term of the fit error matrix. The systematic uncertainty (second uncertainty value) is equal to half of the differences of the determinations obtained fitting Eq. (30) to the points R σ raw +∆R σ raw syst. and R σ raw −∆R σ raw syst. . In the table the χ 2 probability values of the fits performed to the central values are reported.
The values of a obtained analyzing pions and kaons are consistent inside the large uncertainties of about 4%. The value obtained using protons is 14% smaller. The constant term b is about 5% and equal for the three particle beams. Analyses of simulated events produced values of a 10% smaller than the ones obtained using experimental data. The determinations of the constant terms b are 30% larger.
The values of a and b obtained analyzing pion data are consistent within about 2.6 sigmas with the results obtained in a previous study [12].

Summary and conclusions
The results described in this paper were obtained by exposing three modules of the ATLAS Tile Calorimeter to positive pion and kaon and proton beams with energies equal to 16, 18, 20 and 30 GeV and incident at the centre of the front face of a calorimeter module cell with an angle of 14 degrees from the normal. Two Table 13: Values of the parameters a and b obtained fitting Eq. (30) to the experimental and simulated fractional resolution values R σ raw obtained using pions (π), kaons (K) and prtons (p) as a function of 1⁄ E beam [GeV] (see Figure 21). In the case of experimental data results, statistical and systematic uncertainties are reported. Only statistical uncertainties appear in the case of simulated data results. The χ 2 probability values of the fits are reported. Previous pion [12] results are also shown (π old). Cherenkov counters in the beam line made it possible to identify pions, kaons and protons. The effects of electrons contaminating the pion samples in reconstructing the pion energy were determined by exploiting the difference of electromagnetic and hadronic shower profiles in the detector.
The main purpose of the study is to compare the measured energy of the particles with the predictions of the Geant4-based simulation program used in ATLAS to simulated jets produced in proton-proton collisions at the Large Hadron Collider.
Eleven (Nine) determinations of the twelve energy responses (resolutions) normalized to incident beam energy have a total uncertainty smaller than 1.4% (1.9%). In the case of kaons with E beam = 16 GeV, due to the large statistical error, the uncertainty on the determination of R E raw , is equal to 2.4%. The uncertainty values of the determinations of R σ raw obtained in the case of 16 GeV pion and kaon and 18 GeV kaon beams are equal to 3.1%, 20.3% and 10.4% respectively.
Determinations of all the energy responses and of the pion and kaon energy resolutions obtained using experimental and simulated data agree within the uncertainties. The average of the absolute values of the differences of all the energy response measurements was found to be 1.1% with an average total uncertainty of 1.4%. The average difference of all the resolution measurements was found to be 3.4%. The average total un-certainty of pion and kaon (proton) resolution measurements is 5.6% (0.6%).
In the considered E beam range, the measured ratios of the kaon over pion energy responses is constant with a weighted average equal to 0.967 ± 0.002 (-0.014). In parenthesis are reported the differences with the determinations obtained using simulated data. The ratios of the energy responses of protons and pions range between 0.908 ± 0.008 (+0.009) at E beam = 16 GeV to 0.941 ± 0.001 (+0.010) at E beam = 30 GeV. The values of the ratios of the energy resolution determinations are constants. The weighted averages values are R σ raw (K)⁄R σ raw (π) = 0.95 ± 0.01 (-0.011) and R σ raw (p)⁄R σ raw (π) = 0.888 ± 0.005 (+0.011).
The differences of pion, kaon and proton responses and resolutions are due to the different fraction of nonelectromagnetic energy deposited by incident particles: F h (π), F h (K) and F h (p) and to the non-compensating nature of the detector. Data show constant ratios F h (K)/F h (π). The weighted average numerical value is 1.13 ± 0.01 (1.072 ± 0.001). The values in parenthesis were obtained analysing simulated data. The ratio F h (p)/F h (π) decreases from 1.351 ± 0.04 (1.361 ± 0.003) at E beam 16 GeV to 1.24 ± 0.01 (1.281 ± 0.003) at E beam 30 GeV.
As discussed in Section 6.1 the fraction of nonelectromagnetic energy deposited by incident particles can be expressed in terms of the parameters m and E 0 [GeV]. The ratio between the responses to the purely EM and hadronic components of showers e/h describes the non-compensation nature of the calorimeter.. The values of m obtained using experimental (simulated) pions, kaons and protons data are 0.919 ± 0.005 (0.826 ± 0.005), 0.97 ± 0.02 (0.843 ± 0.002) and 0.789 ± 0.003 (0.734 ± 0.002) respectively.