University of Birmingham Longitudinal asymmetry and its effect on pseudorapidity distributions in Pb–Pb collisions at sNN=2.76 TeV

First results on the longitudinal asymmetry and its effect on the pseudorapidity distributions in Pb–Pb collisions at √ s NN = 2.76 TeV at the Large Hadron Collider are obtained with the ALICE detector. The longitudinal asymmetry arises because of an unequal number of participating nucleons from the two colliding nuclei, and is estimated for each event by measuring the energy in the forward neutron-Zero-Degree-Calorimeters (ZNs). The effect of the longitudinal asymmetry is measured on the pseudorapidity distributions of charged particles in the regions | η | < 0 . 9, 2 . 8 < η < 5 . 1 and − 3 . 7 < η < − 1 . 7 by taking the ratio of the pseudorapidity distributions from events corresponding to different regions of asymmetry. The coeﬃcients of a polynomial ﬁt to the ratio characterise the effect of the asymmetry. A Monte Carlo simulation using a Glauber model for the colliding nuclei is tuned to reproduce the spectrum in the ZNs and provides a relation between the measurable longitudinal asymmetry and the shift in the rapidity ( y 0 ) of the participant zone formed by the unequal number of participating nucleons. The dependence of the coeﬃcient of the linear term in the polynomial expansion, c 1 , on the mean value of y 0 is investigated.


Introduction
In a heavy-ion collision, the number of nucleons participating from each of the two colliding nuclei is finite, and will fluctuate event-by-event. The kinematic centre of mass of the participant zone, defined as the overlap region of the colliding nuclei, in general has a finite momentum in the nucleon-nucleon centre of mass frame because of the unequal number of nucleons participating from the two nuclei. This momentum causes a longitudinal asymmetry in the collision and corresponds to a shift of rapidity of the participant zone with respect to the nucleon-nucleon centre of mass (CM) rapidity, termed the rapidity-shift y 0 . The value of y 0 is indicative of the magnitude of the longitudinal asymmetry of the collision [1,2]. Assuming the number of nucleons participating from each of the two nuclei is A and B, the longitudinal asymmetry in participants is defined as α part = A−B A+B and the rapidity-shift can be approximated as y 0 ∼ = 1 2 ln A B at LHC energies [2]. The longitudinal asymmetry may affect the kinematic distribution of particles. The pseudorapidity distribution (dN/dη) of soft particles, averaged over a large number of events, is symmetric in collisions of identical nuclei. These distributions were observed to be asymmetric in collisions of unequal nuclei such as d-Au [3] and p-Pb [4][5][6] and have been explained in terms of the rapidity-shift of the participant zone [7]. In a heavy-ion collision, the effect of the rapidity-shift of the participant zone should be discernible in the distribution of produced particles. This small effect can be estimated by taking the ratio of pseudorapidity distributions in events corresponding to different longitudinal asymmetries [2].
It was suggested that the rapidity distribution of an event, scaled by the average rapidity distribution, can be expanded in terms of Chebyshev polynomials, where the coefficients of expansion are measures of the strength of longitudinal fluctuations and can be determined by measuring the two particle correlation function [8]. Using the same methodology, the event-by-event pseudorapidity distributions are also expanded in terms of Legendre polynomials [9]. The ATLAS collaboration expanded the pseudorapidity distributions in terms of Legendre polynomials and obtained the coefficients by studying pseudorapidity correlations [10].
In the present work, the events are classified according to the asymmetry determined from the measurement of energies of neutron spectators on both sides of the collision [2]. The effect of asymmetry is investigated by taking the ratio of the measured raw dN/dη distributions for events from different regions of the distribution of measured asymmetry. A major advantage of studying this ratio is the cancellation of (i) systematic uncertainties and (ii) the effects of short range correlations. The first measurements of the effect of asymmetry on the raw dN/dη distributions are reported here.
The paper is organised as follows: Sect. 2 provides an introduction to the experimental setup and the details of the data sample. Section 3 discusses the characterisation of the change in raw dN/dη distributions for events classified in different asymmetry regions. Section 4 describes the simulations employed to provide a relation between the measured asymmetry and the rapidity-shift y 0 of the participant zone. The relation between the parameter characterising the change in raw dN/dη distributions is shown for different centralities in Sect. 5, along with its relation to the estimated values of y 0 .

Experimental details and data sample
The analysis uses data from Pb-Pb collision events at √ s NN = 2.76 TeV, recorded in the ALICE experiment in 2010, with a minimum bias trigger [11,12]. A detailed description of the ALICE detectors and their performance can be found elsewhere [13,14].
The event asymmetry is estimated using the energy measured in the two neutron Zero Degree Calorimeters (ZNs). The ZNs were situated 114 metres away from the nominal interaction point (IP) on either side. The ZNs detect only spectator neutrons that are not bound in nuclear fragments, since the latter are bent away by the magnetic field of the LHC separation dipole. The ZN detection probability for neutrons

Analysis and systematic uncertainties
In the present analysis, changes in the raw pseudorapidity distribution of charged particles are investigated for different values of measured asymmetry of the event. The method of measurement of the asymmetry and the parameters characterising the change in dN/dη distributions are discussed in this section.

Analysis
Any event asymmetry due to unequal number of nucleons from the two participating nuclei may manifest itself in the longitudinal distributions, i.e. dN/dy (or dN/dη) of the produced particles because of a shift in the effective CM. Assuming that the rapidity distributions can be described by a Gaussian form [17] with a width σ y and mean y 0 (y 0 = 0.0 for symmetric events), the ratio of the distributions for asymmetric and symmetric events may be written as [2] (dN/dy) asym Here N is the normalisation constant. For any functional form of the rapidity distribution, this ratio may be expanded in a Taylor series. The coefficients c n of the different terms in the expansion depend on the shape and the parameters of the rapidity distribution [2]. In the ALICE experiment, the pseudorapidities of the emitted particles were measured. The effect of a rapidity-shift y 0 on the pseudorapidity distribution is discussed in Sect. 4.2.
The unequal number of participating nucleons will yield a non-zero y 0 of the participant zone and will cause an asymmetry in the number of spectators. This asymmetry can provide information about the mean values of y 0 using the response matrix discussed in Sect. 4. The asymmetry of each event is estimated by measuring the energy in the ZNs on both sides of the interaction vertex: E ZNA on the side referred to as the A-side (η > 0) and E ZNC on the side referred to as the C-side (η < 0). A small difference in the mean and the relative energy resolution of the 1n peak at 1.38 TeV was observed in the performance of the two ZNs [15]. For each centrality interval, the energy distribution in each ZN is divided by its mean, and the width of the E ZNC / E ZNC distribution is scaled to the width of the where ε ZNC(A) is a dimensionless quantity for each event, obtained after scaling the distributions of E ZNC(A) as described above.
For the 15-20% centrality interval, Fig. 1 shows the distribution of the asymmetry α ZN . To investi- Region2 Region3 gate the significance of this distribution, the contribution of the resolution of ZNs to the resolution of the asymmetry parameter α ZN is evaluated. For each centrality interval, values of E ZNC and E ZNA are simulated for each event by assuming a normal distribution peaked at the mean value corresponding to the average number of neutrons and the corresponding energy resolution. The average number of neutrons is estimated by dividing the experimental distribution of energy in ZN by 1.38 TeV. These values are used to obtain α ZN for each event and its distribution. The width of the distribution corresponds to the intrinsic resolution of the measured parameter α ZN and varies from 0.023 to 0.050 from the most peripheral (30-35%) selection to the most central (0-5%) selection. The observed width of 0.13 of the distribution of α ZN reported in Fig. 1 is considerably larger than the resolution of α ZN (0.027 for the centrality interval corresponding to the data in the figure) and the increase in width may be attributed to the event-by-event fluctuations in the number of neutrons detected in each ZN. To investigate the effect of α ZN on the dN/dη distributions, the events are demarcated into three regions of asymmetry by choosing a cut value α cut ZN . These regions correspond to (i) α ZN < −α cut ZN (Region 1), (ii) α ZN ≥ α cut ZN (Region 2) and (iii) −α cut ZN ≤ α ZN < α cut ZN (Region 3). Regions 1 and 2 are referred to as the asymmetric regions and Region 3 is referred to as the symmetric region.
The effect of the measured asymmetry α ZN on the pseudorapidity distributions is investigated by studying the ratio of dN/dη distribution in events from the asymmetric region to those from the symmetric region. There are small differences in the distributions of centrality and vertex position in events of different regions of asymmetry. It is necessary to ensure that any correlation between the ratio of dN/dη and  The ratio of dN/dη for events corresponding to different regions of asymmetry, as shown in Fig. 1, is determined. For |η| < 1.0, the ratio is obtained using dN/dη for tracks. For |η| > 1.0, the ratio shown in Fig. 2(a) and (b) is obtained from amplitudes measured in V0A and the one shown in Fig. 2(c) and (d) is from amplitudes measured in V0C. The squares in Fig. 2 (a) and (c) represent the ratio of dN/dη in the asymmetry Region 1 to that in Region 3 (R13), and the stars represent the corresponding ratio in Region 2 to Region 3 (R23). The filled circles in Fig. 2 (b) and (d) are obtained by (i) reflecting the data points labelled R23 across η = 0 and (ii) taking the averages of R13 and reflected-R23 for |η| < 1.0. A third order polynomial is fitted to the points and the values of the coefficients c n along with the χ 2 are shown. The polynomial fit to the ratio of dN/dη distribution has a dominantly linear term. A small residual detector effect is observed when determining c 1 using data measured in V0A and when using data measured in V0C. In all subsequent discussion, the values of c 1 quoted are the mean of values obtained from the measurements in V0A and V0C.
Considering that the event samples corresponding to different regions of asymmetry are identical in all aspects other than their values of measured α ZN , the observation of non-zero values of c 1 can be attributed to the asymmetry. For a fixed centrality interval, c 1 depends on the choice of α cut ZN . The analysis is repeated for different values of α cut ZN and the dependence of c 1 on α cut ZN is shown in Fig. 3

Systematic uncertainties
The current method of analysis uses the ratio of two dN/dη distributions from events divided on the basis of measurements in ZNs, within a centrality interval. All effects due to limited efficiency, acceptance or contamination would cancel while obtaining the value of the ratio. The contributions to the systematic uncertainties on c 1 are estimated due to the following sources: 1. Centrality selection: the ratio of dN/dη is obtained from the measurements of tracks in the ITS+TPC at midrapidity and charge particle signal amplitudes in the V0 at forward rapidities. For the former, the event centrality is determined using the measurements in the V0 and for the latter using the track multiplicity in the TPC. The analysis is repeated by interchanging the centrality criteria and the resultant change in the values of c 1 for different centrality intervals is in the range 0.1% to 3.6%.
2. V0A and V0C: the systematic uncertainty on the mean value of c 1 is estimated by assuming a uniform probability distribution for the correct value of c 1 to lie between the two values obtained using the charged-particle signal amplitudes measured in the V0A and the V0C. The uncertainty is in the range 2.1% to 4.6% and does not depend on the centrality value.
3. Vertex position: the analysis is repeated for the z position of the interaction vertex |V z | ≤ 3.0 cm.
Longitudinal asymmetry and dN/dη in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration For the most central interval, the results change by less than 0.1%. For the 15-20% centrality interval, the results change by 3.3% and for all other centrality intervals, the changes are less than 1.3%.
4. Weight factors for normalisation: the analysis is also repeated without the weight factors mentioned in Sect. 3.1 for the centrality and the vertex normalisation in the number of events. The change in the results is 4.9% in the most central class and less than 1% for all other centrality intervals.
The total systematic uncertainty is obtained by adding the four uncertainties in quadrature. The resultant uncertainty is in the range 2.3% to 5.8% and is shown by the band in Fig. 8.

Simulations
The simulation used for obtaining a relation between rapidity-shift y 0 and the measurable asymmetry α ZN is described in this section. This simulation has three components: (i) a Glauber Monte Carlo to generate number of participants and spectator protons and neutrons, (ii) a function parametrised to fit the average loss of spectator neutrons due to spectator fragmentation (the loss of spectator neutrons in each event is smeared around this average) and (iii) the response of the ZN to single neutrons. The simulation encompassing the above is referred to in the present work as Tuned Glauber Monte Carlo (TGMC), and reproduces the energy distributions in the ZNs. The effect of y 0 on the pseudorapidity distributions has been estimated using additional simulations for a Gaussian dN/dy and are also described in this section.

Asymmetry and rapidity-shift
The For each generated event, the number of participating protons and neutrons is obtained, enabling a determination of the rapidity-shift y 0 and the various longitudinal asymmetries. If A and B are the number of spectators (spectator neutrons) in the two colliding nuclei, the asymmetry is referred to as α spec (α spec−neut ). Figure 4 (a) shows the correspondence between y 0 and α part . Figures 4 (b) and (c) show the Longitudinal asymmetry and dN/dη in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration relation between y 0 and α spec and α spec−neut respectively [2]. These figures show that the rapidity-shift y 0 can be estimated by measuring α spec or α spec−neut in any experiment that uses Zero Degree Calorimeters. However, the lack of information on the number of participants worsens the precision in determining y 0 . Figure 4 (d) shows the relation between y 0 and α ZN obtained in TGMC, as described in the next paragraph.
The Glauber Monte Carlo is tuned to describe the experimental distributions of ZN energy. For each 1% centrality interval, the mean number of spectator neutrons (N s ) is obtained in the Glauber Monte Carlo. Folding the ZN response yields the simulated values of mean energy as a function of centrality. The experimentally measured mean energy in the ZN is also determined for each 1% centrality interval. The ratio of the measured value of mean energy to the simulated value of mean energy gives the fractional loss (f ) of neutrons due to spectator fragments that veer away due to the magnetic field.  is incorporated to reproduce the experimental distribution of the energy deposited in the ZN shown in Fig. 5 (a). The peak and the RMS of the energy distributions match well. The fractional difference in the position of the peak varies between 3.7% for the 0-5% centrality interval and 0.1% for the 30-35% centrality interval. The fractional difference in RMS for the most central class is 8.6% and is in the range 1.0-2.0% for all other centrality intervals. The distributions of the asymmetry parameter for the TGMC data and the measured data for each centrality interval are shown in Fig. 5 (b). The TGMC contains information of y 0 and α ZN for each event. A scatter plot between y 0 and α ZN is shown in Fig. 4(d) for the 15-20% centrality interval. This constitutes the response matrix. For any measured value of α ZN , the distribution of y 0 can be obtained. Any difference in the experimental and TGMC distributions of α ZN can be accounted for by scaling the y 0 distribution by the ratio of number of events in data to the number in TGMC as with Data (TGMC) in the superscript of number of events, N events , denoting the experimental (TGMC) data. For each of the three regions of asymmetry shown in Fig. 1, corresponding to a chosen value of α cut ZN = 0.1, the distribution of rapidity-shift y 0 obtained using the response matrix is shown in Fig. 6. It is worth mentioning that the width of the distribution of y 0 for events from Region 3, corresponding to −α cut ZN ≤ α ZN < α cut ZN , is comparable to the widths of the corresponding distributions from Regions 1 and

Effect of rapidity-shift on pseudorapidity distributions
The effect of a shift in the rapidity distribution by y 0 on the measurable pseudorapidity distribution (dN/dη) is investigated using simulations. For each event, the rapidity of charged particles is generated from a Gaussian distribution of a chosen width σ y . The pseudorapidity is obtained by using the Blast- . For the most central (0-5%) class, a value 3.86 is used for the width of the rapidity distribution, and a value 4.00 is used for the width of the least central class employed in this analysis (30-35%).
The distribution of rapidity-shift y 0 , similar to the one shown in Fig. 6, is obtained for each centrality interval and each α cut ZN using TGMC. Figure 7 (a) shows the y 0 as a function of α cut ZN for different centralities. One observes a linear relation between the two quantities, showing that an asymmetry in the ZN measurement, arising from the unequal number of participating nucleons, is related to the mean rapidity-shift y 0 . The rapidity distribution of the particles produced in each event is generated assuming a Gaussian form centred about a y 0 , which is generated randomly from the y 0 distribution. Events with a rapidity distribution shifted by y 0 = 0 yield an asymmetric pseudorapidity distribution. A third order polynomial function in η is fitted to the ratio of the simulated dN/dη for the asymmetric region to the simulated dN/dη for the symmetric region. The values of the coefficients in the expansion depend upon the rapidity-shift y 0 and the parameters characterising the distribution [2].
The simulations described above were repeated for different values of α cut ZN to obtain the pseudorapidity distributions for symmetric and asymmetric regions. Fitting third order polynomial functions to the ratios of the simulated pseudorapidity distributions determines the dependence of c 1 on α cut ZN . Figure 7(b) shows that c 1 has a linear dependence on α cut ZN for each centrality interval. The difference in slopes for different centralities is due to differences in the distributions of y 0 and to differences in the widths of the rapidity distributions.
It is important to note that the parameter c 1 , characterising the asymmetry in the pseudorapidty distribution, shows a linear dependence on the parameter α cut ZN in the data sample generated using TGMC and simulations for a Gaussian dN/dy, akin to the dependence of the estimated value of rapidity-shift y 0 for the same sample of events.

Results
The longitudinal asymmetry in a heavy-ion collision has been estimated from the difference in the energy of the spectator neutrons on both sides of the collision vertex. The effect of the longitudinal asymmetry is observed in the ratio of dN/dη distributions corresponding to different asymmetries. The linear term in a polynomial fit to the distribution of the ratio is dominant, and is characterised by its coefficient c 1 . The centrality dependence of the coefficient c 1 for α cut ZN = 0.1 is shown in Fig. 8. It is worth emphasising that the values of c 1 and hence its centrality dependence are affected by (i) the distribution of rapidity-shift y 0 for each centrality interval, (ii) the chosen value of α cut ZN , as seen in Fig. 7 and (iii) the shape or the width of the parent rapidity distribution for each centrality. Figure 8 also shows the results obtained using simulations as described in Sec. 4.2 for α cut ZN = 0.1. The systematic uncertainty on the simulated data sample is estimated by (i) varying the resolution of ZNs from 20% to 30%, (ii) assuming all charged particles are pions and (iii) varying the width of the parent rapidity distribution within the range corresponding to the uncertainties on FWHM quoted in Ref. [20]. The simulated data show a good agreement with the experimental data providing credence to the assumptions of the simulation, in particular that the asymmetry in the distributions arises from the shift of rapidity of the participant zone.
There are two quantities from independent measurements for each selection of asymmetric events. These are (i) c 1 , the parameter characterising the effect of asymmetry in the dN/dη distributions and shown in Fig. 3 and (ii) the mean rapidity-shift y 0 obtained from the measured asymmetry, filtered through the corresponding response matrix (Fig. 4 (d)), and shown in Fig. 7 (a). The relation between c 1 and y 0 is shown in Fig. 9. The parameter c 1 shows a linear dependence on y 0 for each centrality. The difference  in slope is primarily due to different widths of the rapidity distributions for different centralities. If the rapidity distributions were measured instead of dN/dη, then the corresponding parameter c 1 would be Longitudinal asymmetry and dN/dη in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration related to the rapidity-shift as c 1 = y 0 σ y 2 [2].

Conclusions
The present analysis demonstrates the existence of a longitudinal asymmetry in the collision of identical nuclei due to fluctuations in the number of participants from each colliding nucleus. This asymmetry has been measured in the ZNs in the ALICE experiment (Fig. 1), and affects the pseudorapidity distributions, as demonstrated by taking the ratio of distribution of events from the asymmetric region to the corresponding one from the symmetric region (Fig. 2). The effect can be characterised by the coefficient of the linear term in the polynomial expansion of the ratio. The coefficients show a linear dependence on α cut ZN , a parameter to classify the events into symmetric and asymmetric regions (Fig. 3). Different values of α cut ZN correspond to different values of the mean rapidity shift y 0 ( Fig. 7 (a)). The parameter describing the change in the pseudorapidity distributions (c 1 ) has a simple explanation in the rapidity-shift y 0 of the participant zone (Fig. 9). The analysis confirms that the longitudinal distributions are affected by the rapidity-shift of the participant zone with respect to the nucleon-nucleon CM frame. The results provide support to the relevance of number of nucleons affecting the production of charged particles, even at such high energies.
The longitudinal asymmetry is a good variable to classify the events and provides information on the initial state of each event. A systematic study of the effects of longitudinal asymmetry on different observables, e.g. the odd harmonics of anisotropic flow, the forward-backward correlations, the source sizes, in heavy-ion collisions may reveal other characteristics of the initial state and of particle production phenomena. [20] ALICE Collaboration, J. Adam et al., "Centrality evolution of the charged-particle pseudorapidity density over a broad pseudorapidity range in Pb-Pb collisions at √ s NN = 2.76 TeV," Phys. Lett. B754 (2016) 373-385, arXiv:1509.07299 [nucl-ex].