Invariant yield calculation

Chater 6 Invariant yield calculation he invariant yield of the neutral ions and η mesons er one minimum bias collision as a function of the transverse momentum is given by E d3 N d 3 = d 3 N d dydφ = d 2 N 2π d dy, (6.1) where in the last equation isotroic roduction in azimuth is assumed. Using the exerimentally measured quantities the invariant yield is calculated as where: E d3 N d 3 = 1 2π ε vertex Y N trig K trig (1 ε beam ) y Y is the raw yield measured in the bin y; N trig is the number of triggers recorded; 1 1 1 ε acc ε cv Γ γγ /Γ, (6.2) K trig is the trigger rescale factor which is unity for the MinBias events and larger than unity for the Highower data. he roduct N trig K trig then gives the equivalent number of minimum bias events that roduced the yield Y ; ε vertex is the vertex finding efficiency in minimum bias events; ε beam is the beam background contamination in minimum bias events; is the bin for which the yield is calculated; 71

72 CHAPER 6. INVARIAN YIELD CALCULAION y is the raidity range of the measurements, in this analysis y = 1; ε acc is the BEMC accetance and efficiency correction factor; ε cv is a correction for random vetoes; Γ γγ /Γ is the branching ratio of the di-hoton decay channel, equal to 0.988 for π 0 and 0.392 for η [66]. Each of these corrections are described in detail in the following sections. 6.1 Accetance and efficiency correction o calculate the accetance and efficiency correction factor ε acc, a Monte Carlo simulation of the detector was used where neutral ions and their decay hotons were tracked through the SAR detector geometry using GEAN [71]. he simulated signals were assed through the same analysis chain as the real data. he ions were generated in the seudoraidity region 0.3 < η < + 1.3 which is sufficiently large to account for edge effects caused by the calorimeter accetance limits of 0 < η < 1, the azimuth was generated flat in π < φ < +π. he distribution was taken to be flat between zero and 25 GeV/c which amly covers the measured ion range of u to 17 GeV/c. he vertex distribution of the generated ions was taken to be Gaussian in z with a sread of σ = 60 cm and centered at z = 0. he generated ions were allowed to decay into π 0 γγ. he GEAN simulation accounts for all interaction of the decay hotons with the detector, such as air conversion into e + e and showering in the calorimeter or in the material in front. o reroduce a realistic energy resolution of the calorimeter, an additional smearing has to be alied to the energy deosit generated by GEAN in the towers. he effect of this can be seen in Figure 6.1 where the simulated π 0 invariant mass eak is shown in comarison to the + data with and without smearing. An additional sread of 5% was used to reroduce the + data and % for the d + Au data. o reroduce the sectrum of ions in the data, each Monte Carlo event was weighted by a -deendent function. Such weighting technique allows to samle the whole range with good statistical ower while, at the same time, the bin migration effect caused by the finite detector energy resolution is reroduced. A next-to-leading order QCD calculation [72] rovided the initial weight

6.1. ACCEPANCE AND EFFICIENCY CORRECION 73 Figure 6.1: he invariant mass distribution in the real + data (crosses) and in the simulation (histogram). he Monte Carlo roduces a narrower π 0 eak (left) than is observed in the data so that an additional energy smearing was introduced to reroduce the calorimeter resolution (right). function, arametrized as described in Section 7.1, which was subsequently adjusted in an iterative rocedure. As mentioned in Section 4.3, the time deendence of the calorimeter accetance is stored in data tables which are fed into the analysis. In order to reroduce this time deendence in the Monte Carlo, the simulated events were assigned time stams that follow the timeline of the real data taking. In Figure 6.2 is shown, searately for MinBias and Highower data, the accumulated real data statistics er day (histogram) together with the time distribution of the simulated events (full circles). In this way, the geometrical calorimeter accetance (fraction of good towers) was reroduced in the Monte Carlo with a recision of better than 0.5%. In the real data analysis, we use vertices reconstructed from the PC tracks with a sub-millimiter resolution as well as vertices derived from the BBC time of flight measurement with a recision of about 40 cm. o account for this oor resolution, a fraction of the simulated ions had their oint of origin artificially smeared in the z direction. his fraction was taken to be 35% of the generated

74 CHAPER 6. INVARIAN YIELD CALCULAION Events/total 0.2 + MinBias Simulation + Highower Simulation 0.1 0 1 120 130 140 150 160 Day 120 130 140 150 160 Day Figure 6.2: Statistics accumulated er day (histogram) and simulated in the Monte Carlo (full circles), for + MinBias (left) and Highower data (right). ions in case of the + MinBias analysis and taken to be zero for all the other datasets since no BBC vertex was used in these sets (see Chater 4). In Figure 6.3 we show the η and φ distributions of the reconstructed Monte Carlo ions in comarison to the + data. he agreement is satisfactory indicating that the calorimeter accetance is well reroduced in the simulation. In Figure 6.4 the reconstructed of simulated ions is comared to that of ion candidates from the + data. It is seen that the Highower trigger threshold effects are reasonably well reroduced. In Figure 6.5(a) the background subtracted invariant mass distribution is shown in the region 4 < < 5 GeV/c obtained from the + Highower-1 data, together with the corresonding distribution from the Monte Carlo. In order to comare the real and simulated invariant mass distributions for all bins in and for all datasets we have estimated the osition and width of the eaks by Gaussian fits in the eak region. In Figure 6.5(b) are shown the eak ositions obtained from the fit to the + data. It is seen that the eak osition shifts towards higher masses with increasing. his shift is a manifestation of bin migration effects which originate from statistical fluctuations in the calorimeter resonse. Due to the steely falling sectrum the energy resolution will cause a net migration towards larger. Since larger values of imly larger values of M γγ the migration effect will bias the invariant mass eak towards larger values. he good agreement between the data and Monte Carlo indicates that such resolution and migration effects are well reroduced. In Figure 6.5(c) is shown the comarison of the π 0 eak width in data and simulation. he eak width is well reroduced in simulation, which is not surrising since additional smearing was introduced to imrove the comarison

6.1. ACCEPANCE AND EFFICIENCY CORRECION 75 Counts/event 0.07 0.06 0.05 + Highower-1 Simulation 0.04 0.03 0.02 0.01 0 0 0.2 0.4 0.6 0.8 1 η Counts/event 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0-3 -2-1 0 1 2 3 φ [rad] Figure 6.3: Distributions of η (to) and φ (bottom) coordinates of the reconstructed Monte Carlo ions, comared to the ion candidates in the + Highower-1 data. he structure seen in the φ distribution reflects the azimuthal deendence of the calorimeter accetance caused by failing SMD modules. his structure is well reroduced in the simulations.

76 CHAPER 6. INVARIAN YIELD CALCULAION Counts/event -1-2 + MinBias Highower-1 Highower-2 Simulation -3-4 0 2 4 6 8 12 14 16 18 20 Figure 6.4: Distributions of the reconstructed of the Monte Carlo MinBias, Highower-1 and Highower-2 ions, comared to the + data. between data and Monte Carlo, see Figure 6.1. he accetance and efficiency correction factor was calculated from the Monte Carlo simulation as the ratio of the raw yield of neutral ions reconstructed in a bin, to the number of simulated ions with the true in that bin. his was done searately for each trigger using the same ion reconstruction cuts as was done in the real data analysis. In articular, the reconstructed value of seudoraidity was required to fall in the range 0 < η < 1 in both the data and the Monte Carlo while in the latter the generated value of η was also required to fall in this range. In Figure 6.6 are shown the π 0 and η correction factors for all datasets and triggers used in this analysis. he large difference between the MinBias and Highower correction factors is caused by the SMD requirement in the Highower data, while in the MinBias data we accet all reconstructed BEMC oints. he absense of the SMD information also reduces the π 0 reconstruction efficiency at > 3 GeV/c, when the decay hotons are searated by less than two towers. he η reconstruction starts being affected at larger values of.

6.1. ACCEPANCE AND EFFICIENCY CORRECION 77 Counts/max 1.2 1 0.8 (a) + Highower-1 4 < < 5 GeV/c 0.6 0.4 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 [GeV/c 2 ] M γγ Peak osition [GeV/c 2 ] 0.17 0.16 0.15 0.14 (b) MinBias Highower-1 Highower-2 Data MC 0.13 m π 0 Peak width [GeV/c 2 ] 0 (c) 2 4 6 8 12 14 16 0.04 0.03 0.02 0.01 0 0 2 4 6 8 12 14 16 Figure 6.5: Invariant mass sectrum reconstructed in the simulation in comarison to the + Highower-1 data in 4 < < 5 GeV/c bin (to). Peak osition (middle) and width (bottom) in the real data and MC simulation.

78 CHAPER 6. INVARIAN YIELD CALCULAION ε acc 0.4 0.3 + π 0 MinBias Highower-1 Highower-2 d+au π 0 0.2 0.1 1 ε acc 0.3 0.2 0 2+ 4 η6 8 12 14 16 0 d+au 2 4 η6 8 12 14 16 0.1 1 0 0 2 4 6 8 12 14 16 2 4 6 8 12 14 16 Figure 6.6: Accetance and efficiency factor ε acc calculated from the Monte Carlo simulation for the + (left-hand lots) and d+au datasets (right-hand lots). he π 0 and η efficiencies are shown searately in the to and bottom lots, resectively. ε acc 0.14 0.12 0.1 0.08 0.06 0.04 0.02 + Highower-1 π 0 Relaxed SMD cut 0 2 4 6 8 12 14 16 Figure 6.7: Accetance and efficiency correction for the + Highower-1 data, with standard set of cuts and with SMD quality cut removed.

6.2. CORRECIONS FOR RANDOM VEOES 79 We have checked the effect of the SMD quality requirement (at least two adjacent stris in a cluster) on the correction factor for Highower triggered data. In Figure 6.7 is shown the correction factor calculated for the + Highower-1 dataset with (full squares) and without (crosses) the SMD quality requirement. It is seen that this requirement reduces the number of acceted π 0 candidates by about 45%. his exlains the difference between the Highower-1 and Highower-2 (no SMD quality cut) correction factors at large seen in Figure 6.6. o verify a ossible deendence of the accetance correction on the track multilicity and thus on the centrality we have analyzed a samle of generated neutral ions embedded in real d + Au data. hese embedded data are centrally roduced by the SAR offline grou and are used by several analyses in SAR [73]. No significant centrality deendence was found so that same correction factors were alied to the different centrality classes in the d + Au data. 6.2 Corrections for random vetoes his analysis uses the PC as a veto detector to reject charged articles, which introduces false rejection of hoton clusters if an unrelated charged article haens to hit the calorimeter nearby the cluster. In Figure 6.8 we lot the distribution of distances between the BEMC oint and the closest charged track in the event. In this lot one easily distinguishes the eak of real charged articles at small distances, suerimosed on a random comonent which shows u as a shoulder at larger distances. Assuming that the charged tracks are uniformly distributed in η and φ around the BEMC oint it follows that the radial distribution is given by f(d) = D e Dτ, (6.3) where the arameter τ has the meaning of the local track density in the region where the hoton robes it. his arameter is obtained from a simultaneous fit to the data in all bins of the event multilicity M, assuming its linear deendence on the multilicity τ = a + bm. he arametrization (6.3) well describes the random comonent as shown by the full curve in Figure 6.8. he relative amount of random coincidences is then obtained by integrating the fitted curve u to the distance cut and weighting with the multilicity distribution observed in each bin. Searate sets of correction factors were calculated for the different triggers in the + and d + Au data. he results are shown in Figure 6.9 as a function of. We have alied a correction factor of ε cv = 0.94 ± 0.02 to

80 CHAPER 6. INVARIAN YIELD CALCULAION Counts 3 Distance cut 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 D = 2 2 η + φ Figure 6.8: Distribution of the distance between BEMC oints and the closest track, obtained from + Highower-1 data in the bin 4 < < 5 GeV/c. he curve shows a fit to Eq. (6.3) and the vertical line indicates the CPV cut. the + datasets and of ε cv = 0.89 ± 0.02 to the d + Au datasets. he errors assigned to these corrections contribute to a indeendent systematic error on the corrected π 0 and η yields. 6.3 Highower trigger scale factors We have shown in Figure 5.5(c) the distribution of π 0 candidates for the + MinBias, Highower-1 and Highower-2 data. o match the Highower sectra to those of the MinBias a -indeendent scale factor was alied. hese scale factors were estimated as the ratio K trig of observed MinBias to Highower event rates NMB S MB K trig =. (6.4) NH S H Here N MB and N H are the numbers of MinBias and Highower triggers which ass the event selection cuts described in Chater 4. he factors S MB and S H

6.4. VEREX FINDING EFFICIENCY 81 ε cv 1 0.95 0.9 0.85 + MinBias Highower-1 Highower-2 d+au MinBias Highower-1 Highower-2 0.8 0 2 4 6 8 12 14 16 2 4 6 8 12 14 16 Figure 6.9: Charged article veto correction in + (left) and d + Au (right) data. he horizontal line indicates the correction factor ε cv alied to the data while the shaded band corresonds to the systematic uncertainty assigned to the correction factors. are the hardware rescale factors adjusted on a run-by-run basis to accomodate the DAQ bandwidth. In Eq. (6.4), the sums are taken over all runs where both the MinBias and Highower triggers were active. o check the results, the scale factors were also estimated using another method. Here the Highower software filter (see section 4.5.3) was alied to the minimum bias data. he scale factors were then obtained as the ratio of the total number of MinBias events to the number of those which assed the filter. o obtain a more recise Highower-1/Highower-2 relative scale factor, the software filter was alied to the Highower-1 dataset. he results from the two methods agree within 3% for Highower-1 data and within 5% for Highower-2 data. his is taken as the systematic uncertainties on the trigger scale factors. 6.4 Vertex finding efficiency In the π 0 reconstruction it is assumed that the decay hotons originate from the vertex. It is therefore required that each event entering the analysis has a reconstructed vertex. In the + dataset this requirement is always fulfilled because we use the BBC timing information in case the PC vertex reconstruction fails (this haens in about 35% of the minimum bias events). In the d + Au Highower data, the charged track multilicities are large enough to always have a reconstructed PC vertex. However, a PC vertex is missing in about 7% of the minimum bias events and cannot be recovered

82 CHAPER 6. INVARIAN YIELD CALCULAION from BBC information because the BBC is not included in the d + Au minimum bias trigger. Minimum bias events without vertex have low charged track multilicity and the contribution from these very soft events to the π 0 yield above 1 GeV is assumed to be negligible [74]. herefore the correction for vertex inefficiency is alied as a constant normalization factor to the yield and its uncertainty contributes to the total normalization uncertainty of the measured cross sections. he vertex efficiencies were determined to be 0.93 ± 0.01 from a full simulation of d+au minimum bias events as described in [74]. However, this efficiency deends on the centrality and we assume that central events are 0% efficient. Scaling the above efficiency by the ratio of eriheral to total number of d + Au events we obtain an efficiency correction factor of 0.88 ± 0.02 for the samle of eriheral events. Note that the difference between vertex finding efficiencies in MinBias and Highower data is effectively absorbed in the scale factor K trig defined in the revious section. he vertex finding efficiency correction is therefore alied to the minimum bias data as well as to the scaled Highower trigger data. 6.5 Residual beam background contamination he beam background contamination in the d + Au minimum bias trigger has been estimated from an analysis of the RHIC emty bunches to be 5 ± 1 % [75]. In our analysis the beam background in d+au events is rejected when the energy deosit in the calorimeter is much larger than the total energy of all charged tracks reconstructed in the PC, see Section 4.5. o estimate the residual beam background in our data we have analysed a samle of 3 5 minimum bias triggers from unaired RHIC bunches. hese events were assed through the same analysis cuts and reconstruction rocedure as the real data. We observed that about % of the fake triggers assed all cuts and that none of these contained a reconstructed π 0. he residual beam background contamination is thus estimated to be 0.1 5 = 0.5% which is considered to be negligible. In the + data the beam background contamination to the minimum bias trigger rate is also estimated to be negligible due to the BBC coincidence requirement in the trigger and the cut on the BBC vertex osition.

6.6. BIN CENERING SCALE FACORS 83 6.6 Bin centering scale factors o assign a value of to the yield measured in a bin the rocedure from [76] was alied. Here the measured yield, initially lotted at the bin centers, is aroximated by a ower law function of the form f( ) = A (1 + / 0 ) n. (6.5) o each bin a momentum was assigned as calculated from the equation f( ) = 1 f(x)dx. (6.6) he function (6.5) is then re-fitted taking as the abscissa. his rocedure was re-iterated until the values of were stable (tyically after three iterations). Final fitted curves are shown in Figures 6.11 and 6.12. For convenience of comaring results from the various datasets the yields were scaled to the bin centers by the ratio K K = f( ) f( ), (6.7) where is the center of the bin. he statistical and systematic errors were also scaled by the same factor. 6.7 Jacobian correction All calculations in this analysis were erformed in the defined esudoraidity region 0 < η < 1 which corresonds to the raidity region 0 < y < y 0, where the raidity limit y 0 is well aroximated by seudoraidity for a article with momentum much larger than its mass. he correction was alied to account for the raidity limit y 0 being not equal to seudoraidity η = 1, as shown in Figure 6.. his correction is smaller than % for the η data oints at < 3 GeV/c, and is negligible for the other data oints. 6.8 Fully corrected yields he fully corrected π 0 invariant yields er minimum bias event in + and d + Au collisions were calculated from Eq. (6.2) and are shown in the to lots

84 CHAPER 6. INVARIAN YIELD CALCULAION 0 Raidity y 1 0.95 0.9 π 0 η 2 4 6 8 12 14 16 18 Figure 6.: Jacobian correction that accounts for the raidity limit y 0 being not equal to seudoraidity η = 1. of Figure 6.11 and Figure 6.12. he curves in these figures reresent a fit of Eq. (6.5) to the data. In the bottom lots are shown the ratios between the data and the fit. From these lots it is seen that the agreement between the datasets taken with the different triggers is satisfactory. For the calculation of the final cross section results and cross section ratios, the data from three triggers were merged together and only one data oint was chosen in each overlaing bin. he Highower-1 oints were referred over MinBias and Highower-2 over Highower-1 because at high data samles are highly correlated while Highower datasets tyically have smaller statistical error. he systematic uncertainty due to the calorimeter calibration was estimated from δf( ) = df d δ, where δ was taken to be 5% in the d + Au and + data (see Section 4.4), and where the derivative was calculated from the fitted function, Eq. (6.5). his -deendent systematic uncertainty is, on average, 38% in the + data and 44% in the d + Au data.

6.8. FULLY CORRECED YIELDS 85 ] -2 [(GeV/c) N dy 2 d d π 2 1 Invariant yield -1-2 -3-4 -5-6 -7-8 -9 - + MinBias π 0 + Highower-1 π 0 + Highower-2 π 0 Fit to + d+au MinBias π 0 d+au Highower-1 π 0 d+au Highower-2 π 0 Fit to d+au 0 2 4 6 8 12 14 16 18 Invariant yield / Fit 2.5 2 1.5 1 0.5 + MinBias π 0 + Highower-1 π 0 + Highower-2 π 0 Fit to + 0 2 d+au 4 MinBias 6 π8 0 12 14 16 18 2.5 d+au Highower-1 π 0 d+au Highower-2 π 0 2 Fit to d+au 1.5 1 0.5 0 2 4 6 8 12 14 16 18 Figure 6.11: Invariant yield of π 0 er minimum bias event in + and d + Au collisions (to). Curves are the ower law fits given in the text. Invariant yield divided by the fit to the + (middle) and d + Au (bottom) data. he errors shown are statistical only.

86 CHAPER 6. INVARIAN YIELD CALCULAION ] -2 [(GeV/c) N dy 2 d d π 2 1 Invariant yield -1-2 -3-4 -5-6 -7-8 -9 - + MinBias η + Highower-1 η + Highower-2 η Fit to + d+au MinBias η d+au Highower-1 η d+au Highower-2 η Fit to d+au 0 2 4 6 8 12 14 16 18 Invariant yield / Fit 4 2 + MinBias η + Highower-1 η + Highower-2 η Fit to + 0 2 d+au 4 MinBias 6 η 8 12 14 16 18 d+au Highower-1 η d+au Highower-2 η 4 Fit to d+au 2 0 2 4 6 8 12 14 16 18 Figure 6.12: Invariant yield of η meson er minimum bias event in + and d + Au collisions (to). Invariant yield divided by the fit to the + (middle) and d + Au (bottom) data. he errors shown are statistical only.

6.8. FULLY CORRECED YIELDS 87 All systematic error contributions mentioned in this and the revious sections are summarized in able 6.1, classified into the following categories: A oint-by-oint systematic uncertainty; B oint-by-oint -correlated systematic uncertainty, but uncorrelated between datasets; C oint-by-oint -correlated systematic uncertainty, also correlated between datasets; N normalization uncertainty, uncorrelated between datasets. able 6.1: Systematic error contributions. he classifications A, B, C and N are defined in the text. he error contributions to the cross section, the η/π 0 ratio, R CP and R da are indicated in the resective columns. he last column refers to the section where each source of systematic error is described. Source ye E d 3 σ/d 3 η/π 0 R CP R da Section Combinatorial background A + + + + 5.4 Mixed-event background C + + 5.4 Random vetoes N + + + 6.2 Highower scale factors B + + + 6.3 Analysis cuts A + + + + 5.6 Energy scale B + + 6.8 Vertex finding efficiency N + + + 6.4 Min. bias cross section N + + 4.1 Glauber model N coll N + + 4.6