1 Introduction

Quantum chromodynamics (QCD) predicts a transition from hadronic matter to a state of deconfined quarks and gluons, i.e., to the quark-gluon plasma (QGP), at a temperature of \(T_c \approx 150{-}160\,{\mathrm {MeV}}\) at vanishing net baryon number [1, 2]. Energy densities created in Pb–Pb collisions at the LHC are estimated to be sufficiently large to reach this state [3, 4]. At low transverse momenta (roughly \(p_\mathrm{T}\lesssim 3\,{\mathrm {GeV}}/\)c) it is expected that pressure gradients in the QGP produced in an ultrarelativistic collision of two nuclei give rise to a collective, outward-directed velocity profile, resulting in a characteristic modification of hadron spectra [5]. At sufficiently large \(p_\mathrm{T}\) (\(\gtrsim 3{-}8\,{\mathrm {GeV}}\)/c), hadrons in pp and Pb–Pb collisions originate from hard scattering as products of jet fragmentation. Hard-scattered quarks and gluons, produced in the initial stage of the heavy-ion collision, must traverse the QGP that is produced around them and lose energy in the process through interactions with that medium. This phenomenon (“jet quenching”) leads to a modification of hadron yields at high \(p_\mathrm{T}\) [6, 7]. By studying observables related to jet quenching one would like to better understand the mechanism of parton energy loss and to use hard probes as a tool to characterize the QGP.

The modification of the hadron yields for different \(p_\mathrm{T}\) intervals in heavy-ion (A–A) collisions with respect to pp collisions can be quantified with the nuclear modification factor

$$\begin{aligned} R_\mathrm{{AA}}(p_\mathrm{T})=\frac{{\mathrm {d}}^2N/{\mathrm {d}}p_\mathrm{T}{\mathrm {d}}y|_{{\mathrm {AA}}}}{\langle T_{\mathrm{AA}}\rangle \times {\mathrm {d}}^2\sigma /{\mathrm {d}}p_\mathrm{T}{\mathrm {d}}y|_{{\mathrm {pp}}}} \end{aligned}$$

where the nuclear overlap function \(\langle T_{\mathrm{AA}}\rangle \) is related to the average number of inelastic nucleon-nucleon collisions as \(\langle T_{\mathrm{AA}}\rangle = \langle N_{{\mathrm {coll}}} \rangle / \sigma _{{\mathrm {inel}}}^{\mathrm {pp}}\). In the factorization approach of a perturbative QCD calculation of particle production from hard scattering, the overlap function \(T_{\mathrm{AA}}\) can be interpreted as the increase of the parton flux in going from pp to A–A collisions. Without nuclear effects, \(R_\mathrm{{AA}}\) will be unity in the hard scattering regime.

Parton energy loss depends on a number of factors including the transport properties of the medium and its space-time evolution, the initial parton energy, and the parton type [812]. The nuclear modification factor, \(R_\mathrm{{AA}}\), is also affected by the slope of the initial parton transverse momentum spectrum prior to any interaction with the medium and by initial-state effects like the modifications of the parton distributions in nuclei. An important constraint for modeling these effects comes from the study of p–A collisions [13], but also from the study of A–A collisions at different center-of-mass energies (\(\sqrt{s_{{\mathrm {NN}}}}\)) and different centralities. For instance, the increase in \(\sqrt{s_{{\mathrm {NN}}}}\) from RHIC to LHC energies by about a factor 14 results in larger initial energy densities and less steeply falling initial parton spectra [14]. Moreover, at the LHC, pions with \(p_\mathrm{T}\lesssim 50\,{\mathrm {GeV}}/\)c are dominantly produced in the fragmentation of gluons [15], whereas the contribution from quark fragmentation in the same \(p_\mathrm{T}\) region is much larger and more strongly varying with \(p_\mathrm{T}\) at RHIC [16]. Therefore, the pion suppression results at the LHC will be dominated by gluon energy loss, and simpler to interpret than the results from RHIC. Compared to measurements of the \(R_\mathrm{{AA}}\) for inclusive charged hadrons, differences between the baryon and meson \(R_\mathrm{{AA}}\) provide additional information on the parton energy loss mechanism and/or on hadronization in A–A collisions [17, 18]. Experimentally, neutral pions are ideally suited for this as they can be cleanly identified (on a statistical basis) via the decay \(\pi ^0 \rightarrow \gamma \gamma \).

The suppression of neutral pions and charged hadrons at large transverse momentum [1923] and the disappearance of azimuthal back-to-back correlations of charged hadrons in central Au–Au collision at RHIC [24, 25] (see also [2629]) were interpreted in terms of parton energy loss in hot QCD matter. Neutral pions in central Au–Au collisions at \(\sqrt{s_{{\mathrm {NN}}}}= 200\,{\mathrm {GeV}}\) were found to be suppressed by a factor of \(4{-}5\) for \(p_\mathrm{T}\gtrsim 4\,{\mathrm {GeV}}/\)c [30, 31]. The rather weak dependence of \(R_\mathrm{{AA}}\) on \(p_\mathrm{T}\) was described by a large number of jet quenching models [32]. The \(\sqrt{s_{{\mathrm {NN}}}}\) and system size dependence was studied in Cu-Cu collisions at \(\sqrt{s_{{\mathrm {NN}}}}= 19.4\), 62.4, and \(200\,{\mathrm {GeV}}\) [33] and in Au–Au collisions at \(\sqrt{s_{{\mathrm {NN}}}}= 39\), 62.4, and \(200\,{\mathrm {GeV}}\) [22, 34]. In central Cu-Cu collisions the onset of \(R_\mathrm{{AA}}< 1\) was found to occur between \(\sqrt{s_{{\mathrm {NN}}}}= 19.4\) and \(62.4\,{\mathrm {GeV}}\). For unidentified charged hadrons in central Pb–Pb collisions at the LHC, \(R_\mathrm{{AA}}\) was found to increase from \(R_\mathrm{{AA}}< 0.2\) at \(p_\mathrm{T}\approx 7\,{\mathrm {GeV}}/\)c to \(R_\mathrm{{AA}}\approx 0.5\) for \(p_\mathrm{T}\gtrsim 50\,{\mathrm {GeV}}/\)c, in line with a decrease of the relative energy loss with increasing parton \(p_\mathrm{T}\) [3537].

The dependence of the neutral pion \(R_\mathrm{{AA}}\) on \(\sqrt{s_{{\mathrm {NN}}}}\) and \(p_\mathrm{T}\) in Au–Au collisions at RHIC energies for \(2 \lesssim p_\mathrm{T}\lesssim 7\,{\mathrm {GeV}}/\)c is not fully reproduced by jet quenching calculations in the GLV framework which is based on perturbative QCD [34, 38, 39]. This may indicate that, especially for this intermediate \(p_\mathrm{T}\) range, jet quenching calculations do not yet fully capture the relevant physics processes. With the large increase in \(\sqrt{s_{{\mathrm {NN}}}}\) the measurement of \(R_\mathrm{{AA}}\) at the LHC provides a large lever arm to further constrain parton energy loss models. Phenomena affecting pion production in the \(p_\mathrm{T}\) range \(0.6<p_\mathrm{T}< 12\,{\mathrm {GeV}}/\)c of this measurement include collective radial flow at low \(p_\mathrm{T}\) and parton energy loss at high \(p_\mathrm{T}\). The data are therefore well suited to test models aiming at a description of particle production over the full transverse momentum range, including the potentially complicated interplay between jets and the evolving medium.

2 Detector description

Neutral pions were reconstructed via the two-photon decay channel \(\pi ^0 \rightarrow \gamma \gamma \) which has a branching ratio of 98.8 % [40]. Two independent methods of photon detection were employed: with the photon spectrometer (PHOS) which is an electromagnetic calorimeter [41], and with photon conversions measured in the central tracking system using the inner tracking system (ITS) [42] and the time projection chamber (TPC) [43]. In the latter method, referred to as photon conversion method (PCM), conversions out to the middle of the TPC were reconstructed (radial distance \(R \approx 180\,{\mathrm {cm}}\)). The material in this range amounts to \((11.4 \pm 0.5)\) % of a radiation length \(X_0\) for \(|\eta | < 0.9\) corresponding to a plateau value of the photon conversion probability of \((8.6 \pm 0.4)\,\%\). The measurement of neutral pions with two independent methods with different systematics and with momentum resolutions having opposite dependence on momentum provides a valuable check of the systematic uncertainties and facilitates the measurements of neutral pions in a wide momentum range with small systematic uncertainty.

PHOS consists of three modules installed at a distance of \(4.6\,{\mathrm {m}}\) from the interaction point. PHOS subtends \(260^\circ <\varphi <320^\circ \) in azimuth and \(|\eta |<0.13\) in pseudorapidity. Each module has 3584 detection channels in a matrix of \(64\times 56\) cells made of lead tungstate (PbWO\(_4\)) crystals each of size \(2.2\times 2.2 \times 18\) cm\(^3\). The transverse dimensions of the cells are slightly larger than the PbWO\(_4\) Molière radius of \(2\,{\mathrm {cm}}\). The signals from the cells are measured by avalanche photodiodes with a low-noise charge-sensitive preamplifier. In order to increase the light yield and thus to improve energy resolution, PHOS crystals are cooled down to a temperature of \(-25~^\circ \)C. The PHOS cells were calibrated in pp collisions by equalizing the \(\pi ^0\) peak position for all cell combinations registering a hit by a decay photon.

The inner tracking system (ITS) [44] consists of two layers of silicon pixel detectors (SPD) positioned at a radial distance of \(3.9\,{\mathrm {a}}\)nd \(7.6\,{\mathrm {cm}}\), two layers of silicon drift detectors (SDD) at \(15.0\,{\mathrm {a}}\)nd \(23.9\,{\mathrm {cm}}\), and two layers of silicon strip detectors (SSD) at \(38.0\,{\mathrm {a}}\)nd \(43.0\,{\mathrm {cm}}\). The two SPD layers cover a pseudorapidity range of \(|\eta |<2\) and \(|\eta | < 1.4\), respectively. The SDD and the SSD subtend \(|\eta |<0.9\) and \(|\eta |<1.0\), respectively.

The time projection chamber (TPC) [43] is a large (85 m\(^3\)) cylindrical drift detector filled with a Ne/CO\(_2\)/N\(_2\) (85.7/9.5/4.8 %) gas mixture. It covers a pseudorapidity range of \(|\eta |<0.9\) over the full azimuthal angle for the maximum track length of 159 reconstructed space points. With the magnetic field of \(B=0.5\,{\mathrm {T}}\), electron and positron tracks were reconstructed down to transverse momenta of about \(50\,{\mathrm {MeV}}/\)c. In addition, the TPC provides particle identification via the measurement of the specific energy loss (d\(E\)/d\(x\)) with a resolution of 5.5 % [43]. The ITS and the TPC were aligned with respect to each other to a precision better than 100 \(\upmu \)m using tracks from cosmic rays and proton–proton collisions [42].

Two forward scintillator hodoscopes (VZERO-A and VZERO-C) [45] subtending \(2.8 < \eta < 5.1\) and \(-3.7 < \eta < -1.7\), respectively, were used in the minimum bias trigger in the pp and in the Pb–Pb run. The sum of the amplitudes of VZERO-A and VZERO-C served as a measure of centrality in Pb–Pb collisions [46]. Spectator (non-interacting) protons and neutrons were measured with zero degree calorimeters (ZDCs), located close to the beam pipe, \(114\,{\mathrm {m}}\) away from the interaction point on either side of the ALICE detector [44].

3 Data processing

3.1 Event selection

The pp sample at \(\sqrt{s}=2.76\) TeV was collected in the 2011 LHC run. The minimum bias trigger (MB\(_{{\mathrm {OR}}}\)) in the pp run required a hit in either VZERO hodoscope or a hit in the SPD. Based on a van der Meer scan the cross section for inelastic pp collisions was determined to be \(\sigma _{{\mathrm {inel}}} = (62.8^{+2.4}_{-4.0} \pm 1.2)\,{\mathrm {mb}}\) and the MB\(_{{\mathrm {OR}}}\) trigger had an efficiency of \(\sigma _{{\mathrm {MB_{OR}}}}/\sigma _{{\mathrm {inel}}} = 0.881^{+0.059}_{-0.035}\) [47]. The results were obtained from samples of \(34.7 \times 10^6\) (PHOS) and \(58 \times 10^6\) (PCM) minimum bias pp collisions corresponding to an integrated luminosity \(\mathcal{L}_\mathrm{int} = 0.63~\text{ nb }^{-1}\) and \(\mathcal{L}_\mathrm{int} = 1.05~\text{ nb }^{-1}\), respectively. PHOS and the central tracking detectors used in the PCM were in different readout partitions of the ALICE experiment which resulted in the different integrated luminosities.

The Pb–Pb data at \(\sqrt{s_{{\mathrm {NN}}}}= 2.76\,{\mathrm {TeV}}\) were recorded in the 2010 LHC run. At the ALICE interaction region up to 114 bunches, each containing about \(7 \times 10^7\) \(^{208}\)Pb ions, were collided. The rate of hadronic interactions was about 100 Hz, corresponding to a luminosity of about \(1.3 \times 10^{25}\,{\mathrm {cm^{-2} s^{-1}}}\). The detector readout was triggered by the LHC bunch-crossing signal and a minimum bias interaction trigger based on trigger signals from VZERO-A, VZERO-C, and SPD [46]. The efficiency for triggering on a hadronic Pb–Pb collision ranged between 98.4 and 99.7 %, depending on the minimum bias trigger configuration. For the centrality range 0-80 % studied in the Pb–Pb analyses \(16.1 \times 10^6\) events in the PHOS analysis and \(13.2 \times 10^6\) events in the PCM analysis passed the offline event selection.

In both pp and Pb–Pb analyses, the event selection was based on VZERO timing information and on the correlation between TPC tracks and hits in the SPD to reject background events coming from parasitic beam interactions. In addition, an energy deposit in the ZDCs of at least three standard deviations above the single-neutron peak was required for Pb–Pb collisions to further suppress electromagnetic interactions [46]. Only events with a reconstructed vertex in \(| z_{{\mathrm {vtx}}} | < 10\,{\mathrm {cm}}\) with respect to the nominal interaction vertex position along the beam direction were used.

3.2 Neutral pion reconstruction

The PHOS and PCM analyses presented here are based on methods previously used in pp collisions at \(\sqrt{s} = 0.9\) and \(7\,{\mathrm {TeV}}\) [48]. Neutral pions were reconstructed using the \(\pi ^0\rightarrow \gamma \gamma \) decay channel either with both photon candidates detected in PHOS or both photons converted into \(e^+e^-\) pairs and reconstructed in the central tracking system. For the photon measurement with PHOS adjacent lead tungstate cells with energy signals above a threshold (\(12\,{\mathrm {MeV}}\)) were grouped into clusters [49]. The energies of the cells in a cluster were summed up to determine the photon energy. The selection of the photon candidates in PHOS was different for pp and Pb–Pb collisions due to the large difference in detector occupancy. For pp collisions cluster overlap is negligible and combinatorial background small. Therefore, only relatively loose photon identification cuts on the cluster parameters were used in order to maximize the \(\pi ^0\) reconstruction efficiency: the cluster energy for pp collisions was required to be above the minimum ionizing energy \(E_{{\mathrm {cluster}}}>0.3\) GeV and the number of cells in a cluster was required to be greater than two to reduce the contribution of hadronic clusters. In the case of the most central Pb–Pb collisions about 80 clusters are reconstructed in PHOS, resulting in an occupancy of up to 1/5 of the 10,752 PHOS cells. This leads to a sizable probability of cluster overlap and to a high combinatorial background in the two-cluster invariant mass spectra. A local cluster maximum was defined as a cell with a signal at least \(30\,{\mathrm {MeV}}\) higher than the signal in each surrounding cell. A cluster with more than one local maximum was unfolded to several contributing clusters [49]. As the lateral width of showers resulting from hadrons is typically larger than the one of photon showers, non-photonic background was reduced by a \(p_\mathrm{T}\) dependent shower shape cut. This cut is based on the eigenvalues \(\lambda _0\), \(\lambda _1\) of the covariance matrix built from the cell coordinates and weights \(w_i=\max [0,w_0+\log (E_i/E_{{\mathrm {cluster}}})]\), \(w_0=4.5\) where \(E_i\) is the energy measured in cell \(i\). In the Pb–Pb case only cells with a distance to the cluster center of \(R_{{\mathrm {disp}}}=4.5\,{\mathrm {cm}}\) were used in the dispersion calculation. A 2D \(p_\mathrm{T}\)-dependent cut in the \(\lambda _0\)-\(\lambda _1\) plane was tuned to have an efficiency of \({\sim }0.95\) using pp data. In addition, clusters associated with a charged particle were rejected by application of a cut on the minimum distance from a PHOS cluster to the extrapolation of reconstructed tracks to the PHOS surface [49]. This distance cut depended on track momentum and was tuned by using real data to minimize false rejection of photon clusters. The corresponding loss of the \(\pi ^0\) yield was about 1 % in pp collisions (independent of \(p_\mathrm{T}\)). In Pb–Pb collisions the \(\pi ^0\) inefficiency due to the charged particle rejection is about 1 % in peripheral and increases to about 7 % in central Pb–Pb collisions. In addition, to reduce the effect of cluster overlap, the cluster energy was taken as the core energy of the cluster, summing over cells with centers within a radius \(R_{{\mathrm {core}}}=3.5\,{\mathrm {cm}}\) of the cluster center of gravity, rather than summing over all cells of the cluster. By using the core energy, the centrality dependence of the width and position of the \(\pi ^0\) peak is reduced, due to a reduction of overlap effects. The use of the core energy leads to an additional non-linearity due to energy leakage outside \(R_{{\mathrm {core}}}\): the difference between full and core energy is negligible at \(E_{{\mathrm {cluster}}} \lesssim 1\,{\mathrm {GeV}}\) and reaches \({\sim } 4\) % at \(E_{{\mathrm {cluster}}} \sim 10\,{\mathrm {GeV}}\). This non-linearity, however, is well reproduced in the GEANT3 Monte Carlo simulations [50] of the PHOS detector response (compare \(p_\mathrm{T}\) dependences of peak positions in data and Monte Carlo in Fig. 2) and is corrected for in the final spectra.

PHOS is sensitive to pile-up from multiple events that occur within the \(6\,{\mathrm {\upmu s}}\) readout interval of the PHOS front-end electronics. The shortest time interval between two bunch crossings in pp collisions was \(525\,{\mathrm {ns}}\). To suppress photons produced in other bunch crossings, a cut on arrival time \(|t|<265\,{\mathrm {ns}}\) was applied to reconstructed clusters which removed 16 % of the clusters. In the Pb–Pb collisions, the shortest time interval between bunch crossing was \(500\,{\mathrm {ns}}\), but the interaction probability per bunch crossing was much smaller than in pp collisions. To check for a contribution from other bunch crossings to the measured spectra, a timing cut was applied, and the pile-up contribution was found to be negligible in all centrality classes. Therefore, a timing cut was not applied in the final PHOS Pb–Pb analysis.

Fig. 1
figure 1

(Color online) Invariant mass spectra in selected \(p_\mathrm{T}\) slices for PCM (upper row) and PHOS (lower row) in the \(\pi ^0\) mass region for pp (left column), \(60{-}80\,\%\) (middle column) and \(0{-}10\,\%\) (right column) Pb–Pb collisions. The histogram and the filled points show the data before and after background subtraction, respectively. For the \(0{-}10\,\%\) class the invariant mass distributions after background subtraction were scaled by a factor 15 and 5 for PCM and PHOS, respectively, for better visibility of the peak. The positions and widths of the \(\pi ^0\) peaks were determined from the fits, shown as blue curves, to the invariant mass spectra after background subtraction

The starting point of the conversion analysis is a sample of photon candidates corresponding to track pairs reconstructed by a secondary vertex (V0) finding algorithm [49, 51]. In this step, no constraints on the reconstructed invariant mass and pointing of the momentum vector to the collision vertex were applied. Both tracks of a V0 were required to contain reconstructed clusters (i.e., space points) in the TPC. V0’s were accepted as photon candidates if the ratio of the number of reconstructed TPC clusters over the number of findable clusters (taking into account track length, spatial location, and momentum) was larger than 0.6 for both tracks. In order to reject \(K_s^0\), \(\Lambda \), and \(\bar{\Lambda }\) decays, electron selection and pion rejection cuts were applied. V0’s used as photon candidates were required to have tracks with a specific energy loss in the TPC within a band of [\(-3 \sigma \), \(5 \sigma \)] around the average electron d\(E\)/d\(x\), and of more than \(3\sigma \) above the average pion d\(E\)/d\(x\) (where the second condition was only applied for tracks with measured momenta \(p > 0.4\,{\mathrm {GeV}}/\)c). Moreover, tracks with an associated signal in the TOF detector were only accepted as photon candidates if they were consistent with the electron hypothesis within a \(\pm 5 \sigma \) band. A generic particle decay model based on the Kalman filter method [52] was fitted to a reconstructed V0 assuming that the particle originated from the primary vertex and had a mass \(M_{V0}=0\). Remaining contamination in the photon sample was reduced by cutting on the \(\chi ^2\) of this fit. Furthermore, the transverse momentum \(q_T = p_e \sin \theta _{V0,e}\) [53] of the electron, \(p_e\), with respect to the V0 momentum was restricted to \(q_T < 0.05\,{\mathrm {GeV}}/\)c. As the photon is massless, the difference \(\Delta \theta = |\theta _{e^-} - \theta _{e^+}|\) of the polar angles of the electron and the positron from a photon conversion is small and the bending of the tracks in the magnetic field only results in a difference \(\Delta \varphi = |\varphi _{e^-} - \varphi _{e^+}|\) of the azimuthal angles of the two momentum vectors. Therefore, remaining random track combinations, reconstructed as a V0, were suppressed further by a cut on the ratio of \(\Delta \theta \) to the total opening angle of the \(e^+e^-\) pair calculated after propagating both the electron and the positron \(50\,{\mathrm {cm}}\) from the conversion point in the radial direction. In order to reject \(e^+e^-\) pairs from Dalitz decays the distance between the nominal interaction point and the reconstructed conversion point of a photon candidate had to be larger than \(5\,{\mathrm {cm}}\) in radial direction. The maximum allowed radial distance for reconstructed V0’s was \(180\,{\mathrm {cm}}\).

Pile-up of neutral pions coming from bunch crossings other than the triggered one also has an effect on the PCM measurement. At the level of reconstructed photons, this background is largest for photons for which both the electron and the positron were reconstructed with the TPC alone without tracking information from the ITS. These photons, which typically converted at large radii \(R\), constitute a significant fraction of the total PCM photon sample, which is about \(67\,\%\) in case of the pp analysis. This sample is affected because the TPC drift velocity of \(2.7\,{\mathrm {cm/\upmu s}}\) corresponds to a drift distance of \(1.41\,{\mathrm {cm}}\) between two bunch crossings in the pp run which is a relatively short distance compared to the width of \(\sigma _z \approx 5\,{\mathrm {cm}}\) of the distribution of the primary vertex in the \(z\) direction. The distribution of the distance of closest approach in the \(z\) direction (\({\mathrm {DCA}}_z\)) of the straight line defined by the reconstructed photon momentum is wider for photons from bunch crossings other than the triggered one. The \({\mathrm {DCA}}_z\) distribution of photons which had an invariant mass in the \(\pi ^0\) mass range along with a second photon was measured for each \(p_\mathrm{T}\) interval. Entries in the tails at large \({\mathrm {DCA}}_z\) were used to determine the background distribution and to correct the neutral pion yields for inter bunch pile-up. For the pp analysis, this was a \(5{-}7\,\%\) correction for \(p_\mathrm{T}\gtrsim 2\,{\mathrm {GeV}}/\)c and a correction of up to \(15\,\%\) at lower \(p_\mathrm{T}\) (\(p_\mathrm{T}\approx 1\,{\mathrm {GeV}}/\)c). In the Pb–Pb case the correction at low \(p_\mathrm{T}\) was about 10 %, and became smaller for higher \(p_\mathrm{T}\) and for more central collisions. For the \(20{-}40\,\%\) centrality class and more central classes the pile-up contribution was negligible and no pile-up correction was applied. In the PCM as well as in the PHOS analysis, events for which two or more pp or Pb–Pb interactions occurred in the same bunch crossing were rejected based on the number of primary vertices reconstructed with the SPD [49] which has an integration time of less than \(200\,{\mathrm {ns}}\).

In the PHOS as well as in the PCM analysis, the neutral pion yield was extracted from a peak above a combinatorial background in the two-photon invariant mass spectrum. Examples of invariant mass spectra, in the \(\pi ^0\) mass region, are shown in Fig. 1 for selected \(p_\mathrm{T}\) bins for pp collisions, and peripheral and central Pb–Pb collisions. The combinatorial background was determined by mixing photon candidates from different events. In the PCM measurement the combinatorial background was reduced by cutting on the energy asymmetry \(\alpha = |E_{\gamma _1}-E_{\gamma _2}|/(E_{\gamma _1}+E_{\gamma _2})\), where \(\alpha < 0.65\) was required for the central classes (\(0{-}5\), \(5{-}10\), \(10{-}20\), \(20{-}40~\%\)) and \(\alpha < 0.8\) for the two peripheral classes (\(40{-}60\), \(60{-}80\,\%\)). In both analyses the mixed-event background distributions were normalized to the right and left sides of the \(\pi ^0\) peak. A residual correlated background was taken into account using a linear or second order polynomial fit. The \(\pi ^0\) peak parameters were obtained by fitting a function, Gaussian or a Crystal Ball function [54] in the PHOS case or a Gaussian combined with an exponential low mass tail to account for bremsstrahlung [55] in the PCM case, to the background-subtracted invariant mass distribution, see Fig. 1. The Crystal Ball function was used in the PHOS analysis of pp data. A Gaussian was used alternatively to determine systematic uncertainties of the peak parameters. In the Pb–Pb case with worse resolution and smaller signal/background ratios, the difference between Crystal Ball and Gaussian fits disappeared and only the latter were used in the PHOS analysis. In the case of PHOS the number of reconstructed \(\pi ^0\)’s was obtained in each \(p_\mathrm{T}\) bin by integrating the background subtracted peak within 3 standard deviations around the mean value of the \(\pi ^0\) peak position. In the PCM analysis, the integration window was chosen to be asymmetric (\(m_{\pi ^0}-0.035\) GeV/\(\mathrm{c}^2\), \(m_{\pi ^0}+0.010\) GeV/\(\mathrm{c}^2\)) to take into account the left side tail of the \(\pi ^0\) peak due to bremsstrahlung energy loss of electrons and positrons from photon conversions. In both analyses the normalization and integration windows were varied to estimate the related systematic uncertainties. The peak positions and widths from the two analyses are compared to GEANT3 Monte Carlo simulations in Fig. 2 as a function of \(p_\mathrm{T}\). The input for the GEANT3 simulation came from the event generators PYTHIA 8 [56] and PHOJET [57] in the case of pp collisions (with roughly equal number of events) and from HIJING [58] in the case of Pb–Pb collisions. For the PCM analysis the full width at half maximum (FWHM) divided by \(2\sqrt{2\ln 2} \approx 2.35\) is shown. Note the decrease of the measured peak position with \(p_\mathrm{T}\) in Pb–Pb collisions for PHOS. This is due to the use of the core energy instead of the full cluster energy. At low \(p_\mathrm{T}\) in central Pb–Pb collisions, shower overlaps can increase the cluster energy thereby resulting in peak positions above the nominal \(\pi ^0\) mass. A good agreement in peak position and width between data and simulation is observed in both analyses. The remaining small deviations in the case of PHOS were taken into account as a systematic uncertainty related to the global energy scale.

Fig. 2
figure 2

(Color online) Reconstructed \(\pi ^0\) peak width (upper row) and position (lower row) as a function of \(p_\mathrm{T}\) in pp collisions at \(\sqrt{s}=2.76\,{\mathrm {TeV}}\) (a, d), peripheral (b, e) and central (c, f) Pb–Pb collisions at \(\sqrt{s_{{\mathrm {NN}}}}= 2.76\,{\mathrm {TeV}}\) in PHOS and in the photon conversion method (PCM) compared to Monte Carlo (MC) simulations. The horizontal line in (df) indicates the nominal \(\pi ^0\) mass

The correction factor \(\varepsilon (p_\mathrm{T})\) for the PHOS detector response and the acceptance \(A(p_\mathrm{T})\) were calculated with GEANT3 Monte Carlo simulations tuned to reproduce the detector response. The factor \(\varepsilon (p_\mathrm{T})\) takes the loss of neutral pions due to analysis cuts, effects of the finite energy resolution and, in case of Pb–Pb collisions, effects of shower overlaps into account. The shape of the \(\pi ^0\) input spectrum needed for the calculation of \(\varepsilon (p_\mathrm{T})\) was determined iteratively by using a fit of the corrected spectrum of a given pass as input to the next. In the case of Pb–Pb collisions the embedding technique was used in the PHOS analysis: the PHOS response to single \(\pi ^0\)’s was simulated, the simulated \(\pi ^0\) event was added to a real Pb–Pb event on the cell signal level, after which the standard reconstruction procedure was performed. The correction factor \(\varepsilon (p_\mathrm{T}) = (N_{{\mathrm {rec}}}^{\mathrm {after}}(p_\mathrm{T})-N_{{\mathrm {rec}}}^{\mathrm {before}}(p_\mathrm{T}))/N_{{\mathrm {sim}}}(p_\mathrm{T})\) was defined as the ratio of the difference of the number of reconstructed \(\pi ^0\)’s after and before the embedding to the number of simulated \(\pi ^0\)’s. In the pp case, the PHOS occupancy was so low that embedding was not needed and \(\varepsilon (p_\mathrm{T})\) was obtained from the \(\pi ^0\) simulations alone. Both in the Pb–Pb and the pp analysis, an additional 2 % channel-by-channel decalibration was introduced to the Monte Carlo simulations, as well as an energy non-linearity observed in real data at low energies which is not reproduced by the GEANT simulations. This non-linearity is equal to \(2.2\,\%\) at \(p_\mathrm{T}=1\) GeV/\(c\) and decreases rapidly with \(p_\mathrm{T}\) (less than \(0.5\,\%\) at \(p_\mathrm{T}>3\) GeV/\(c\)). For PHOS, the \(\pi ^0\) acceptance \(A\) is zero for \(p_\mathrm{T}<0.4\) GeV/\(c\). The product \(\varepsilon \cdot A\) increases with \(p_\mathrm{T}\) and saturates at about \(1.4\times 10^{-2}\) for a neutral pion with \(p_\mathrm{T}>15\) GeV/\(c\). At high transverse momenta (\(p_\mathrm{T}>25\) GeV/\(c\)) \(\varepsilon \) decreases because of merging of clusters of \(\pi ^0\) decay photons due to the decreasing average opening angle of the \(\pi ^0\) decay photons. The correction factor \(\varepsilon \) does not show a centrality dependence for events in the \(20{-}80\) % class, but in the most central bin it increases by \(\sim \!10\) % due to an increase in cluster energies caused by cluster overlap.

Table 1 Summary of the relative systematic uncertainties in percent for selected \(p_\mathrm{T}\) bins for the PHOS and the PCM analyses

In the PCM, the photon conversion probability of about 8.6 % is compensated by the large TPC acceptance. Neutral pions were reconstructed in the rapidity interval \(|y|<0.6\) and the decay photons were required to satisfy \(|\eta | < 0.65\). The \(\pi ^0\) efficiency increases with \(p_\mathrm{T}\) below \(p_\mathrm{T}\approx 4\,{\mathrm {GeV}}/\)c and remains approximately constant for higher \(p_\mathrm{T}\) at values between \(1.0 \times 10^{-3}\) in central collisions (\(0{-}5\,\%\), energy asymmetry cut \(\alpha < 0.65\)) and \(1.5 \times 10^{-3}\) in peripheral collisions (\(60{-}80\,\%\), \(\alpha < 0.8\)). For the centrality classes \(0{-}5\), \(5{-}10\), \(10{-}20\), \(20{-}40\,\%\), for which \(\alpha < 0.65\) was used, the \(\pi ^0\) efficiency varies between \(1.0 \times 10^{-3}\) and \(1.2 \times 10^{-3}\). This small centrality dependence is dominated by the centrality dependence of the V0 finding efficiency. Further information on the PHOS and PCM efficiency corrections can be found in [49].

The invariant differential neutral pion yield was calculated as

$$\begin{aligned} E \frac{\mathrm{d}^3 N }{\mathrm{d}^3p} = \frac{1}{2\pi } \frac{1}{N_\mathrm{events}} \frac{1}{p_\mathrm{T}} \frac{1}{\varepsilon \,A}\frac{1}{ Br }\frac{N^{\pi ^0}}{\Delta y \Delta p_\mathrm{T}}, \end{aligned}$$

where \(N_\mathrm{events}\) is the number of events; \(p_\mathrm{T}\) is the transverse momentum within the bin to which the cross section has been assigned after the correction for the finite bin width \(\Delta p_\mathrm{T}\), \( Br \) is the branching ratio of the decay \(\pi ^0\rightarrow \gamma \gamma \), and \(N^{\pi ^0}\) is the number of reconstructed \(\pi ^0\)’s in a given \(\Delta y\) and \(\Delta p_\mathrm{T}\) bin. Finally, the invariant yields were corrected for the finite \(p_\mathrm{T}\) bin width following the prescription in [59], i.e., by plotting the measured average yield at a \(p_\mathrm{T}\) position for which the differential invariant yield coincides with the bin average. Secondary \(\pi ^0\)’s from weak decays or hadronic interactions in the detector material were subtracted using Monte Carlo simulations. The contribution of \(\pi ^0\)’s from K\(^0_{{\mathrm {s}}}\) as obtained from the used event generators was scaled in order to reproduce the measured K\(^0_{{\mathrm {s}}}\) yields [60]. The correction for secondary \(\pi ^0\)’s was smaller than \(2\,\%\) (\(5\,\%\)) for \(p_\mathrm{T}\gtrsim 2\,{\mathrm {GeV}}/\)c in the pp as well as in the Pb–Pb analysis for PCM (PHOS).

A summary of the systematic uncertainties for two representative \(p_\mathrm{T}\) values in pp, peripheral and central Pb–Pb collisions is shown in Table 1. In PHOS, one of the largest sources of the systematic uncertainty both at low and high \(p_\mathrm{T}\) is the raw yield extraction. It was estimated by varying the fitting range and the assumption about the shape of the background under the peak. In central collisions, major contributions to the systematic uncertainty are due to the efficiency of photon identification and the global energy scale. The former was evaluated by comparing efficiency-corrected \(\pi ^0\) yields, calculated with different identification criteria. The latter was estimated by varying the global energy scale within the tolerance which would still allow to reproduce the peak position in central and peripheral collisions. The uncertainty related to the non-linearity of the PHOS energy response was estimated by introducing different non-linearities into the Monte Carlo simulations under the condition that the simulated \(p_\mathrm{T}\) dependence of the \(\pi ^0\) peak position and peak width was still consistent with the data. The uncertainty of the PHOS measurement coming from the uncertainty of the fraction of photons lost due to conversion was estimated by comparing measurements without magnetic field to the measurements with magnetic field.

In the PCM measurement, the main sources of systematic uncertainties include the knowled