Hasan Abdulrahman

Baptiste Magnier

Philippe Montesinos

A New Objective Supervised Edge Detection Assessment using Hysteresis Thresholds

Keywords: edge detection, supervised evaluation, hysteresis

Useful for the visual perception of a human, edge detection remains a crucial stage in numerous image processing applications. Therefore, one of the most challenging goals in contour extraction is to operate algorithms that can process visual information as humans need. Hence, to ensure that it is reliable, an edge detection technique needs to be severely assessed before being used it in a computer vision tools. To achieve this task, a supervised evaluation computes a score between a ground truth edge map and a candidate image. Theoretically, by varying the hysteresis thresholds of the thin edges, the minimum score of the measure corresponds to the best edge map, compared to the ground truth. In this study, a new supervised edge map quality measure is proposed, where the minimum score of the measure is associated with an edge map in which the main structures of the desired objects are distinctive.

Introduction on edge detection and thresholding

Edge detection is an important field in image processing because this process frequently attempts to capture the most important structures in the image. Hence, edge detection represents a fundamental step concerning computer vision approaches. Furthermore, edge detection itself could be used to qualify a region segmentation technique. Additionally, the edge detection assessment remains very useful in image segmentation, registration, reconstruction or interpretation. Hence, it is hard to design an edge detector which is able to extract the exact edge with good localization and orientation from an image. In the literature, different techniques have emerged and, due to its importance, edge detection continues to be an active research area [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF]. The best-known and useful edge detection methods are based on gradient computing first-order fixed operators [START_REF] Sobel | Camera Models and Machine Perception[END_REF][START_REF] Canny | A computational approach to edge detection[END_REF]. Oriented operators compute the maximum energy in an orientation [START_REF] Freeman | The design and use of steerable filters[END_REF][START_REF] Jacob | Design of steerable filters for feature detection using Canny-like criteria[END_REF][START_REF] Geusebroek | Fast anisotropic Gauss filtering[END_REF] or two directions [START_REF] Magnier | Fast anisotropic edge detection using Gamma correction in color images[END_REF]. Typically, these methods are composed of three steps:

The final step remains a difficult stage in image processing, however it represents a crucial operation to compare several segmentation algorithms. In edge detection, the hysteresis process uses the connectivity information of the pixels belonging to thin contours and thus remains a more elaborated method than binary thresholding. Simply, this technique determines a contour image that has been thresholded at different levels (low: τ L and high: τ H ). The low threshold τ L determines which pixels are considered as edge points if at least one point higher than τ H exists in a contour chain where all the pixel values are also higher than τ L , as represented with a signal in Fig. 1. Thus, the lower the thresholds are, the more the undesirable pixels are preserved.

Usually, in order to compare several edge detection methods, the user has to try some thresholds to select the ones that appear visually as the best edge maps in quality. However, this assessment suffers from a main drawback: segmentations are compared using the threshold (deliberately) chosen by the user, this evaluation is very subjective and not reproducible. Hence, the purpose is to use the dissimilarity measures without any user intervention for an objective assessment. Finally, to consider a valuable edge detection assessment, the evaluation process should produce a result that correlates with the perceived quality of the edge image, which relies on human judgment [START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF][START_REF] Abdulrahman | From contours to ground truth: How to evaluate edge detectors by filtering[END_REF][START_REF] Heath | A robust visual method for assessing the relative performance of edge-detection algorithms[END_REF]. In other words, a reliable edge map should characterize all the relevant structures of an image as closely as possible, without any disappearance of desired contours. Nevertheless, a minimum of spurious pixels can be created by the edge detector, disturbing at the same time the visibility of the main/desired objects to detect.

In this paper, a novel technique is presented to compare edge detection techniques by using hysteresis thresholds in a supervised way, being consistent with the visual perception of a human. Indeed, by comparing a ground truth contour map with an ideal edge map, several assessments can be compared by varying the parameters of the hysteresis thresholds. This study shows the importance to penalize stronger the false negative points, compared to the false positive points, leading to a new edge detection evaluation algorithm. The experiment using synthetic and real images demonstrated that the proposed method obtains contours maps closer to the ground truth without requiring tuning parameters and outperforms other assessment methods in an objective way.

Type of operator Fixed operator [START_REF] Sobel | Camera Models and Machine Perception[END_REF][START_REF] Canny | A computational approach to edge detection[END_REF] Oriented Filters [START_REF] Freeman | The design and use of steerable filters[END_REF][START_REF] Jacob | Design of steerable filters for feature detection using Canny-like criteria[END_REF][START_REF] Geusebroek | Fast anisotropic Gauss filtering[END_REF] Half Gaussian Kernels [START_REF] Magnier | Fast anisotropic edge detection using Gamma correction in color images[END_REF] Gradient magnitude A supervised evaluation criterion computes a dissimilarity measure between a segmentation result and a ground truth obtained from synthetic data or an expert judgment (i.e. manual segmentation) [11][12][13] [START_REF] Abdulrahman | A new normalized supervised edge detection evaluation[END_REF]. In this paper, the closer to 0 the score of the evaluation is, the more the segmentation is qualified as good. This work focusses on comparisons of supervised edge detection evaluations and proposes a new measure, aiming at an objective assessment.

|∇I| = I 2 0 + I 2 π/2 |∇I| = max θ∈[0,π[ |Iθ| |∇I| = max θ∈[0,2π[ Iθ -min θ∈[0,2π[ Iθ Gradient direction η = arctan I π/2 I0 η = arg max θ∈[0,π[ |Iθ| + π 2 η = arg max θ∈[0,2π[ Iθ + arg min θ∈[0,2π[ Iθ /2

Error measures involving only statistics

To assess an edge detector, the confusion matrix remains a cornerstone in boundary detection evaluation methods. Let G t be the reference contour map corresponding to ground truth and D c the detected contour map of an original image I. Comparing pixel per pixel G t and D c , the 1st criterion to be assessed is the common presence of edge/non-edge points. A basic evaluation is composed of statistics; to that end, G t and D c are combined. Afterwards, denoting | • | as the cardinality of a set, all points are divided into four sets (see Fig. 3): Several edge detection evaluations involving confusion matrix are presented in Table 1. Computing only FPs and FNs [START_REF] Magnier | Fast anisotropic edge detection using Gamma correction in color images[END_REF] or their sum enables a segmentation assessment to be performed. The complemented Performance measure P * m considers directly and simultaneously the three entities T P , F P and F N to assess a binary image and decreases with improved quality of detection.

Another way to display evaluations is to create Receiver Operating Characteristic (ROC) [START_REF] Bowyer | Edge detector evaluation using empirical ROC curves[END_REF] curves or Precision-Recall (PR) [START_REF] Martin | Learning to detect natural image boundaries using local brightness, color, and texture cues[END_REF], involving True Positive Rates (T P R) and False Positive Rates (F P R): T P R = T P T P +F N and F P R = F P F P +T N . Derived from T P R and F P R, the three measures Φ, χ 2 and F α (detailed in Table 1) are frequently used. The complement of these measures enables to translate a value close to 0 as a good segmentation.

These measures evaluate the comparison of two edge images, pixel per pixel, tending to severely penalize a (even slightly) misplaced contour, as illustrated in Fig. 2. Consequently, some evaluations resulting from the confusion matrix recommend incorporating spatial tolerance. Tolerating a distance from the true contour and integrating several TPs for one detected contour can penalize efficient edge detection methods, or, on the contrary, advantage poor ones (especially for corners or small objects). Thus, from the discussion below, the assessment should penalize a misplaced edge point proportionally to the distance from its true location (some examples in [START_REF] Abdulrahman | A new normalized supervised edge detection evaluation[END_REF], and, as shown in Fig. 2).

P * m (Gt, Dc) = 1 - T P |Gt ∪ Dc| = 1 - T P T P + F P + F N Complemented Φ measure [16] Φ * (Gt, Dc) = 1 -T P R • T N T N +F P Complemented χ 2 measure [17] χ 2 * (Gt, Dc) = 1 -T P R-T P -F P 1-T P -F P • T P +F P +F P R T P +F P Complemented Fα measure [18] F * α (Gt, Dc) = 1 - P REC • T P R α • T P R+(1-α)•P REC , with P REC = T P T P +F P and α ∈]0; 1]
Table 2. List of error measures involving distances, generally: k = 1 or k = 2.

Error measure name Formulation Parameters

Pratt's FoM [START_REF] Abdou | Quantitative design and evaluation of enhancement/thresholding edge detectors[END_REF] F oM (Gt,

Dc) = 1 - 1 max (|Gt| , |Dc|) • p∈Dc 1 1 + κ • d 2 Gt (p) κ ∈ ]0; 1] F oM revisited [21] F (Gt, Dc) = 1 - 1 |Gt|+β•F P • p∈Gt 1 1 + κ • d 2 Dc (p) κ ∈ ]0; 1] and β ∈ R +
Combination of F oM and statistics [START_REF] Boaventura | Method to evaluate the performance of edge detector[END_REF] d4

(Gt, Dc) = 1 2 • (T P -max (|Gt| , |Dc|)) 2 + F N 2 + F P 2 (max (|Gt| , |Dc|)) 2 + F oM (Gt, Dc) κ ∈ ]0; 1] and β ∈ R + Symmetric FoM [14] SF oM (Gt, Dc) = 1 2 • F oM (Gt, Dc) + 1 2 • F oM (Dc, Gt) κ ∈ ]0; 1] Maximum FoM [14] M F oM (Gt, Dc) = max (F oM (Gt, Dc) , F oM (Dc, Gt)) κ ∈ ]0; 1] Yasnoff measure [23] Υ (Gt, Dc) = 100 |I| • p∈Dc d 2 Gt (p) None Hausdorff distance [24] H (Gt, Dc) = max max p∈Dc (dG t (p)), max p∈Gt (dD c (p)) None Maximum distance [11] f2d6 (Gt, Dc) = max 1 |Dc| • p∈Dc dG t (p), 1 |Gt| • p∈Gt dD c (p) None Distance to Gt [25][13] D k (Gt, Dc) = 1 |Dc| • k p∈Dc d k Gt (p), k = 1 for [25] k ∈ R + Oversegmentation [26] Θ (Gt, Dc) = 1 F P • p∈Dc dG t (p) δT H k for [26]: k ∈ R + and δT H ∈ R * + Undersegmentation [26] Ω (Gt, Dc) = 1 F N • p∈Gt dD c (p) δT H k for [26]: k ∈ R + and δT H ∈ R * + Symmetric distance [11][13] S k (Gt, Dc) = k p∈Dc d k Gt (p)) + p∈Gt d k Dc (p) |Dc ∪ Gt| , k = 1 for [11] k ∈ R + Baddeley's Delta Met- ric [27] ∆ k (Gt, Dc) = k 1 |I| • p∈I |w(dG t (p)) -w(dD c (p))| k k ∈ R + and a convex function w : R → R Edge map quality measure [28] Dp (Gt, Dc) = 1/2 |I|-|Gt| • p∈Dc 1 - 1 1 + α•d 2 Gt (p) + 1/2 |Gt| • p∈Gt 1 - 1 1 + α•d 2 Gt∩Dc (p) α ∈ ]0; 1] Magnier et al. measure [29] Γ (Gt, Dc) = F P +F N |Gt| 2 • p∈Dc d 2 Gt (p) None
Complete distance measure [START_REF] Abdulrahman | A new normalized supervised edge detection evaluation[END_REF] Ψ (Gt, Dc)

= F P +F N |Gt| 2 • p∈Gt d 2 Dc (p) + p∈Dc d 2 Gt (p) None

Assessment involving distances of misplaced pixels

A reference-based edge map quality measure requires that a displaced edge should be penalized in function not only of FPs and/or FNs but also of the distance from the position where it should be located. Table 2 of FPs and FNs [START_REF] Abdulrahman | From contours to ground truth: How to evaluate edge detectors by filtering[END_REF]. A perfect segmentation using an over-segmentation measure could be an image including no edge points and an image having most undesirable edge points (FPs) concerning under-segmentation evaluations (see Fig. 3). Also, another limitation of only over-and under-segmentation evaluations are that several binary images can produce the same result (Fig. 2). Therefore, as demonstrated in [START_REF] Abdulrahman | From contours to ground truth: How to evaluate edge detectors by filtering[END_REF], a complete and optimum edge detection evaluation measure should combine assessments of both over-and under-segmentation. Among the distance measures between two contours, one of the most popular descriptors is named the Figure of Merit (F oM ). Nonetheless, for F oM , the distance of the FNs is not recorded and are strongly penalized as statistic measures (see above). For example, in Fig. 3, F oM (G t , C) > F oM (G t , M ), whereas M contains both FPs and FNs and C only FNs. Further, for the extreme cases:

(b) M (g) G t vsM (c) C (h) G t vsC (d) T (i) G t vsT (e) B (
• if F P = 0: F oM (Gt, Dc) = 1 -T P/|Gt| = 1 -(|Gt| -F N )/|Gt|, • if F N = 0: F oM (Gt, Dc) = 1 - 1 max(|G t |,|Dc|) • p∈Dc∩¬G t 1 1+κ•d 2 G t (p) .
When F N >0 and F P constant, it behaves like matrix-based error assessments (Fig. 2). Moreover, for F P >0, the F oM penalizes the over-detection very low compared to the under-detection. On the contrary, the F measure computes the distances of FNs but not of the FPs, so F behaves inversely to F oM . Also, d 4 measure depends particularly on T P , F P , F N and F oM but penalizes FNs like the F oM measure. SF oM and M F oM take into account both distances of FNs and FPs, so they can compute a global evaluation of a contour image. However, M F oM does not consider FPs and FNs at the same time, contrary to SF oM . Another way to compute a global measure is presented in [START_REF] Panetta | A new reference-based edge map quality measure[END_REF] with the edge map quality measure D p . The right term computes the distances of the FNs between the closest correctly detected edge pixel, i.e. G t ∩ D c . Finally, D p is more sensitive to FNs than FPs because of the coefficient 1 |I|-|Gt| . A second measure widely computed in matching techniques is represented by the Hausdorff distance H, which measures the mismatch of two sets of points [START_REF] Huttenlocher | A multi-resolution technique for comparing images using the Hausdorff distance[END_REF]. This max-min distance could be strongly deviated by only one pixel which can be positioned sufficiently far from the pattern (Fig. 3). To improve the measure, one idea is to compute H with a proportion of the maximum distances; let us note H 15% this measure for 15% of the values [START_REF] Huttenlocher | A multi-resolution technique for comparing images using the Hausdorff distance[END_REF]. Nevertheless, as pointed out in [START_REF] Dubuisson | A modified Hausdorff distance for object matching[END_REF], an average distance from the edge pixels in the candidate image to those in the ground truth is more appropriate, like S k or Ψ . Eventually, Delta Metric (∆ k ) [START_REF] Baddeley | An error metric for binary images[END_REF] intends to estimate the dissimilarity between each element of two binary images, but is highly sensitive to distances of misplaced points [START_REF] Abdulrahman | From contours to ground truth: How to evaluate edge detectors by filtering[END_REF] [START_REF] Abdulrahman | A new normalized supervised edge detection evaluation[END_REF].

A new objective edge detection assessment measure: In [START_REF] Abdulrahman | A new normalized supervised edge detection evaluation[END_REF] a measure of the edge detection assessment is developed: it is denoted Ψ (Tab. 2) and improvemes the over-segmentation measure Γ , by combining both d Gt and d Dc , see Fig. 3. Ψ gives the same weight for d Gt and d Dc in its assessment of errors. Thus, using Ψ , a missing edge remains not enough penalized contrary to the distance of FPs which could be too important. Another example, in Fig. 3, Ψ (G t , C) < Ψ (G t , T ) whereas C must be more penalized because of FNs which does not allow to identify the object (also Fig. 5). The solution proposed here is to penalize stronger the distances of the FNs depending on the number of TPs:

λ(Gt, Dc) = F P + F N |Gt| 2 • p∈Dc d 2 G t (p) + min |Gt| 2 , |Gt| 2 T P 2 • p∈G t d 2 Dc (p) (1)
The term influencing the penalization of FN distances can be rewritten as:

|Gt| 2 T P 2 = F N +T P T P 2 = 1 + F N T P 2
1, ensuring a stronger penalty for d 2 Dc , compared to d 2

Gt . When T P = 0, the min function avoids the multiplication by infinity; moreover, the number of FNs is large, corresponding to a strong penalty with the weight term |G t | 2 (see Fig. 4 left). When |G t | = T P , λ is equivalent to Ψ and Γ (see Fig. 3, image T ). Also, compared to Ψ , λ penalizes more D c having FNs, than D c with only FPs, as illustrated in Fig. 3 (images C andT). Finally, the weight |Gt| 2 T P 2 tunes the λ measure by considering an edge map of better quality when FNs points are localized close to the desired contours D c .

The next subsection details the way to evaluate an edge detector in an objective way. Results presented in this communication show the importance to penalize stronger the false negative points, compared to the false positive points because the desired objects are not always completely visible by using ill-suited evaluation measure, and, λ provides a reliable edge detection assessment. Dissimilarity measures are used for an objective assessment using binary images. Instead of choosing manually a threshold to obtain a binary image (see Fig. 3 in [START_REF] Abdulrahman | From contours to ground truth: How to evaluate edge detectors by filtering[END_REF]), the purpose is to compute the minimal value of a dissimilarity measure by varying the thresholds (τ L , τ H ) of the thin edges (see Table 1). Thus, compared to a ground truth contour map, the ideal edge map for a measure corresponds to the desired contour at which the evaluation obtains the minimum score for the considered measure among the thresholded (binary) images. Theoretically, this score corresponds to the thresholds at which the edge detection represents the best edge map, compared to the ground truth contour map [START_REF] Fernández-Garcıa | Characterization of empirical discrepancy evaluation measures[END_REF][START_REF] Chabrier | Comparative study of contour detection evaluation criteria based on dissimilarity measures[END_REF][START_REF] Abdulrahman | From contours to ground truth: How to evaluate edge detectors by filtering[END_REF]. Fig. 4 right illustrates the choice of a contour map in function of τ L and τ H . Since small thresholds lead to heavy over-segmentation and strong thresholds may create numerous false negative pixels, the minimum score of an edge detection evaluation should be a compromise between under-and over-segmentation (detailed in [START_REF] Abdulrahman | From contours to ground truth: How to evaluate edge detectors by filtering[END_REF]).

As demonstrated in [START_REF] Abdulrahman | From contours to ground truth: How to evaluate edge detectors by filtering[END_REF], the significance of the ground truth map choice influences on the dissimilarity evaluations. Indeed, if not reliable [START_REF] Hou | Boundary detection benchmarking: Beyond F-measures[END_REF], an inaccurate ground truth contour map in terms of localization penalizes precise edge detectors and/or advantages the rough algorithms as edge maps presented in [START_REF] Heath | A robust visual method for assessing the relative performance of edge-detection algorithms[END_REF][START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF]. For these reasons, the ground truth edge map concerning the real image in our experiments is built in a semi-automatic way detailed in [START_REF] Abdulrahman | From contours to ground truth: How to evaluate edge detectors by filtering[END_REF].

Experimental results

In these experiments, the importance of an assessment to penalize stronger the false negative points is enlightened, compared to the false positive points. In order to study the performance of the contour detection evaluation measures, the hysteresis thresholds vary and the minimum score of the studied measure corresponds to the best edge map. The thin edges of both synthetic and real noisy images are computed by five or six edge detectors: Sobel [START_REF] Sobel | Camera Models and Machine Perception[END_REF], Canny [START_REF] Canny | A computational approach to edge detection[END_REF], Steerable Filters of order 1 (SF 1 ) [START_REF] Freeman | The design and use of steerable filters[END_REF] or 5 (SF 5 ) [START_REF] Jacob | Design of steerable filters for feature detection using Canny-like criteria[END_REF], Anisotropic Gaussian Kernels (AGK) [START_REF] Geusebroek | Fast anisotropic Gauss filtering[END_REF] and Half Gaussian Kernels (H-K) [START_REF] Magnier | Fast anisotropic edge detection using Gamma correction in color images[END_REF]. Fig. 5 presents the results for 14 measures with their associated scores (bars) according to the hysteresis parameters. In the one hand, we must take into account the obtained edge map, and on the other hand the measure score. Generally, the optimal edge map for F oM , SF oM , f 2 d 6 , Ψ and λ measures allows to distinct the majority of the desired edges for each contour detection operator (except Sobel), whereas for the other assessments, contours are too disturbed by undesirable points or distinguished with high difficulty (especially Ψ which does not penalizes enough FNs). Note that SF oM measure does not classify the Sobel algorithm as less efficient. Concerning the experiment with a real image in Fig. 6, 8 measures are compared together. For F oM , H, ∆ k and S k , the ideal edge maps concerning Sobel edge detector are highly corrupted by undesirable contours, the main objects are not recognizable. The other segmentations are also disturbed by undesirable pixels for F oM , H and ∆ k . Moreover, the higher score for ∆ k (AGK) does not represent the more disturbed map. Ultimately, using λ, the essential structures are visible in the optimal contour map for each edge detector (objects are easily recognizable).

Moreover, contrary to H, F oM , d 4 , ∆ k and S k measures, the scores of λ are coherent, in relation to the obtained segmentations (Sobel and H-K results).

Conclusion and future works

This study presents a new supervised edge detection assessment method λ which enables to assess a contour map in an objective way. Based on the theory of the dissimilarity evaluation measures, the objective evaluation allows to evaluate 1st-order edge detectors. Indeed, the segmentation which obtains the minimum score of a measure is considered as the best one. Theory and experiments prove that the minimum score of the new dissimilarity measure λ corresponds to the best edge quality map evaluations, which is similarly closer to the ground truth, compared to the other methods. On the one hand, this new measure takes into account the distances of false positive points, in the other hand, it considers the distance of false negative points tuned by a weight. This weight depends on the number of false negative points: the more it is elevated, the more the segmentation is penalized. Thus, this enables to obtain objectively an edge map containing the main structures, similar to the ground truth, concerning a reliable edge detector. Finally, the computation of the minimum score of a measure does not require tuning parameters, which represents a huge advantage. For this purpose, we plan in a future study to deeply compare the robustness of several edge detection algorithms and use the new measure in object recognition.

Fig. 1 .

 1 Fig.1. Gradient magnitude and orientation computation for a scalar image I and example of hysteresis threshold applied along a contour chain. I θ represents the image derivative using a first-order filter at the θ orientation (in radians).

•

  True Positive points (TPs), common points of Gt and Dc: T P = |Dc ∩ Gt|, • False Positive points (FPs), spurious detected edges of Dc: F P = |Dc ∩ ¬Gt|, • False Negative points (FNs), missing boundary points of Dc: F N = |¬Dc ∩ Gt|, • True Negative points (TNs), common non-edge points: T N = |¬Dc ∩ ¬Gt|.

  reviews the most relevant measures involving distances. Thus, for a pixel p belonging to the desired contour D c , d Gt (p) represents the minimal Euclidian distance between p and G t . If p belongs to the ground truth G t , d Dc (p) is the minimal distance between p and D c . On the one hand, some distance measures are specified in the evaluation of over-segmentation (i.e. presence of FPs), like: Υ , D k , Θ and Γ . On the other hand, Ω measure assesses an edge detection by computing only an under segmentation (FNs). Other edge detection evaluation measures consider both distances

  (a) Gt (b) D1 (c) D2 F * α (Gt, D1)=1.000 F * α (Gt, D2)=1.000 Υ (Gt, D1)=13.223 Υ (Gt, D2)=13.223 F oM ((Gt, D1)= 0.3939 F oM (Gt, D2)=0.3939 H(Gt, D1)=1.4142 H(Gt, D2)= 5.3852 S k k=2 (Gt, D1)=1.0414 S k k=2 (Gt, D2)=1.6993 λ(Gt, D1)=0.4482 λ(Gt, D2)=0.5725

Fig. 2 .

 2 Fig. 2. Different Dc: FPs and number of FNs are the same for D1 and for D2.

Fig. 3 .

 3 Fig. 3. Results of evaluation measures and images for the experiments.

Fig. 4 .VII 2 . 3

 423 Fig. 4. Number of FNs penalizes λ and computation of a measure minimum score.

Fig. 5 . 6 .1 1 MeasureFig. 6 .

 5616 Fig. 5. Comparison of best maps and minimum scores for different evaluation measures. The bars legend is presented in Fig. 6. Gt and original image are available in Fig.3.

Table 1 .

 1 List of error measures involving only statistics.

P erf ormance measure

[START_REF] Grigorescu | Contour detection based on nonclassical receptive field inhibition[END_REF]

Acknowledgements

The authors thank the Iraqi Ministry of Higher Education and Scientific Research for funding and supporting this work and reviewers for their remarks.


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C Deplano

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M A Diaz Corchero

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Ø Djuvsland

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D Domenicis Gimenez

B Dönigus

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F M Fionda

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S Foertsch

P Foka

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N Grion

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F Guber

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T Gunji

A Gupta

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I B Guzman

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G Hamar

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A Harton

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D Hatzifotiadou

S Hayashi

S T Heckel

E Hellbär

H Helstrup

A Herghelegiu

G Herreracorral

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B A Hess

K F Hetland

H Hillemanns

C Hills

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D Horak

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P Hristov

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T J Humanic

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S A Iga Buitron

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M Irfan

V Isakov

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B Jacak

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S Jadlovska

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C Jahnke

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M A Janik

P H S Y Jayarathna

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M Kim

Charged-particle multiplicity distributions over a wide pseudorapidity range in proton-proton collisions at √ s = 0.9, 7, and 8 TeV

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Introduction

The multiplicity of charged particles produced in high-energy pp collisions is one of the key observables to describe the global properties of the interactions and has been the subject of long standing experimental and theoretical investigations. The pp multiplicity distributions of primary-charged particles have been measured for five increasingly wider pseudorapidity ranges. A primary-charged particle is a charged particle with a mean proper lifetime τ larger than 1 cm/c, which is either produced directly in the interaction, or from decays of particles with τ smaller than 1 cm/c, excluding particles produced in interactions with material [1]. The results are determined using both the Silicon Pixel Detector (SPD) and the Forward Multiplicity Detector (FMD) in ALICE to widen the pseudorapidity coverage with respect to previous ALICE results [START_REF] Aamodt | Charged-particle multiplicity measurement in proton-proton collisions at √ s = 0.9 and 2[END_REF][START_REF] Aamodt | Charged-particle multiplicity measurement in proton-proton collisions at √ s = 7 TeV with ALICE at LHC[END_REF][START_REF] Adam | Charged-particle multiplicities in proton-proton collisions at √ s = 0.9 to 8 TeV[END_REF], which made exclusive use of the SPD. The extension of the pseudorapidity coverage allows us to increase the high-multiplicity reach of the distributions by around 70-90% with respect to the previous ALICE publication [START_REF] Adam | Charged-particle multiplicities in proton-proton collisions at √ s = 0.9 to 8 TeV[END_REF], exploring a wider phase space.

The multiplicity distribution of charged particles produced in high-energy pp collisions is sensitive to the number of interactions between quarks and gluons contained in the proe-mail: alice-publications@cern.ch tons and to underlying mechanisms of particle production. At LHC energies, the particle production is dominated by soft QCD processes, which cannot be treated perturbatively and can only be modeled phenomenologically. On the other hand, as the colliding energy grows, the particle production receives increased contributions from hard scattering, which can be treated perturbatively.

We have compared directly our data to previous measurements from CMS [START_REF] Khachatryan | Charged particle multiplicities in pp interactions at √ s = 0.9, 2.36, and 7 TeV[END_REF]. ATLAS and LHCb use different p T and η ranges [START_REF] Aad | Charged-particle multiplicities in pp interactions measured with the ATLAS detector at the LHC[END_REF][START_REF] Aaij | Measurement of charged particle multiplicities and densities in pp collisions at √ s = 7 TeV in the forward region[END_REF], making the direct comparison impossible. This manuscript presents also an overview of the parameters obtained when fitting multiplicity distributions with the sum of two Negative Binomial Distributions (NBDs). Additionally, the results have been compared to simulations regularly used at LHC [START_REF] Skands | Tuning Monte Carlo generators: the perugia tunes[END_REF][START_REF] Bopp | Rapidity gaps and the PHOJET Monte Carlo[END_REF][START_REF] Sjstrand | An introduction to PYTHIA 8.2[END_REF][START_REF] Pierog | EPOS LHC: test of collective hadronization with data measured at the CERN large hadron collider[END_REF] and calculations based on saturation density of gluons in the colliding hadrons [START_REF] Schenke | Multiplicity distributions in p+p, p+A and A+A collisions from Yang-Mills dynamics[END_REF][START_REF] Mclerran | Intrinsic fluctuations of the proton saturation momentum scale in high multiplicity p+p collisions[END_REF].

The manuscript is organized in the following way: Sect. 2 describes the detectors used to measure the charged-particle multiplicity distributions. Section 3 explains the analysis procedure in detail. The systematic uncertainties are described in Sect. [START_REF] Adam | Charged-particle multiplicities in proton-proton collisions at √ s = 0.9 to 8 TeV[END_REF] and the results along with comparisons to models are presented in Sect. 5, which contains also the analysis of the NBD fits. A brief summary and conclusions are finally given in Sect. 6.

Experimental setup

Full details of the sub-detectors are given elsewhere [START_REF] Aamodt | The ALICE experiment at the CERN LHC[END_REF]. ALICE is designed to measure particles over a wide kinematic range -3.4 < η < 5.0. Only the sub-detectors used in this analysis are described, namely the V0 scintillation counters, the SPD, and the FMD.

V0 detector

The V0 detector [START_REF] Abbas | Performance of the ALICE VZERO system[END_REF] is composed of two arrays of 32 scintillators positioned at 330 cm (V0-A) and -90 cm (V0-C) from the nominal interaction point (IP) along the beam axis. Each array has a ring structure segmented into 4 radial and 8 azimuthal sectors. The detector has full azimuthal coverage in the pseudorapidity ranges 2.8 < η < 5.1 and -3.7 < η < -1.7. The signal amplitudes and times are recorded for each of the 64 scintillators. The V0 is appropriate for triggering, thanks to the good timing resolution of each scintillator (1 ns) along with its large acceptance for detecting charged particles. Different V0 trigger settings are used in the analysis.

Silicon pixel detector

The SPD are the two innermost cylindrical layers of the ALICE Inner Tracking System (ITS) [START_REF] Aamodt | The ALICE experiment at the CERN LHC[END_REF] surrounding the beam line. The layers have full azimuthal coverage and radii of 3.9 and 7.6 cm with 9.8 × 10 6 silicon diodes, each of size 50 × 425 µm 2 . The first layer of the SPD has the largest pseudorapidity coverage of the ITS (|η| < 1.98 for collisions at the nominal IP). Besides the readout of individual pixels with signals above a certain threshold, each SPD chip provides a fast signal every 100 ns, indicating a presence of fired pixels (FastOR), making it suitable for triggering. Charged particles can deposit energy in more than one pixel of the SPD. The offline reconstruction combines such neighboring signals into a a single cluster. The charged-particle multiplicity can then be estimated by counting the number of clusters detected in a SPD layer. This analysis uses only clusters from the inner layer of the SPD to provide the largest pseudorapidity coverage for particle detection. Alternatively, clusters from the two SPD layers together with the primary vertex can be combined to form tracklets [START_REF] Adam | Charged-particle multiplicities in proton-proton collisions at √ s = 0.9 to 8 TeV[END_REF], allowing to select primary particles with very high efficiency. The charged-particle multiplicity is then estimated by counting the number of tracklets.

Forward multiplicity detector

The purpose of the FMD is to extend, with high spatial resolution, the charged-particle detection acceptance beyond the reach of the SPD and central detectors in ALICE [START_REF] Cortese | ALICE technical design report on forward detectors: FMD, T0 and V0[END_REF]. The FMD is a silicon strip detector and consists of three subdetectors placed at 320 cm (FMD1), 79 cm (FMD2), and -69 cm (FMD3) from the nominal IP along the beam pipe. FMD2 and FMD3 contain both inner and outer rings of silicon strips. FMD1 is located farther from the IP and has only one inner ring. Inner rings consist of 10 sensors, each with two azimuthal sectors and 512 strips with radii from 4.2 to 17.2 cm. Outer rings contain 20 sensors each again with two azimuthal sectors, but with 256 strips, with radii from 15.4 to 28.4 cm. Each ring (inner or outer), therefore, contains 10,240 strips giving in total 51,200 strips. The FMD has full azimuthal coverage in the pseudorapidity ranges -3.4 < η < -1.7 and 1.7 < η < 5.0.

The FMD records, for each strip, the energy deposited by charged particles traversing the detector. Various selection criteria, see [START_REF] Abbas | Centrality dependence of the pseudorapidity density distribution for charged particles in Pb-Pb collisions at √ s NN = 2.76 TeV[END_REF][START_REF] Adam | Centrality evolution of the charged-particle pseudorapidity density over a broad pseudorapidity range in Pb-Pb collisions at √ s NN = 2.76 TeV[END_REF] for details, are applied to the energy measured in each strip to determine if the signal corresponds to a single particle traversing only this strip or also a neighboring strip. The number of particles traversing the FMD is determined taking into account only the signals which pass these selection criteria. The majority of the particles which reach the FMD, however, are secondary particles produced in interactions with the beam pipe, the material of the ITS, cables and support structures [START_REF] Abbas | Centrality dependence of the pseudorapidity density distribution for charged particles in Pb-Pb collisions at √ s NN = 2.76 TeV[END_REF]. Therefore, a detailed Monte Carlo simulation is needed to determine the number of primary particles produced in the collision.

Analysis procedure

The multiplicity distribution of the primary particles is affected by many detector effects, such as dead detector regions and secondary particle production. These detector effects must be minimized and corrected for as they have increasing effects when determining accurately the probability of progressively higher-multiplicity events. The unfolding method is used to correct for the detector effects, as will be described in the following.

Event selection

Collisions at three different center of mass energies (0.9, 7, and 8 TeV) are analyzed. The data used for the analysis were collected at low beam currents and low pileup during three data taking periods: the 0.9 and 7 TeV samples were acquired in 2010, while the 8 TeV sample was collected in 2012. The last sample is the most affected by the pileup contamination. For this reason we selected few specific runs with low contamination from pileup events for this energy, and used data taken with interaction rate not exceeding 1 kHz. A pileup is defined as more than one collision occurring during the readout time of the detector (300 ns, for the SPD, and 2 µs sampling time, for the FMD). Such events produce a bias towards larger multiplicity that enhance mostly the tail of the multiplicity distribution. Table 1 shows the number of selected events at each energy and the average number of interactions per bunch crossing, μ , measured by the experiment [START_REF] Abelev | Performance of the ALICE Experiment at the CERN LHC[END_REF]. This parameter is determined experimentally and for this measurements, in which μ 1, the average probability of having more than one interaction in a single bunch crossing, where at least one interaction occurs, is around 1 -2% ( μ /2).

Inelastic non-diffractive scatterings are the dominant processes in pp collisions, for which most of the hadrons are produced as a consequence of an exchange of color charge. On the contrary, diffractive events can be single-, double-, or [START_REF] Collins | An Introduction to Regge Theory and High-Energy Physics[END_REF], diffraction occurs when the Pomeron interacts with the proton and produces a system of particles, called the diffractive system. The case in which only one of the protons dissociates is called singlediffractive.

The signals of the V0 and SPD are used to select events where at least one interaction occurred, which are triggered by requiring the detection of at least one particle in either the V0-A, V0-C, or SPD (MB OR ). Events are divided into three classes depending on further requirements. The first class includes all inelastic events (the INEL class), which is the same condition as used to select events where an interaction occurred (MB OR trigger). The second class (the INEL > 0) requires the presence of at least one charged particle (tracklet) in the region |η| < 1.0 in addition to the INEL condition. This class has higher trigger efficiency and, therefore, reduced corrections relative to the INEL event class. The third class requires charged particles to be detected in both the V0-A and the V0-C (MB AND ). This class is used to remove the majority of the single-diffractive events and is, therefore, called the non-single-diffractive (NSD) event class.

To remove interactions of the beam with residual gas in the beam pipe, further selection criteria are applied to the event sample. Since these interactions can occur anywhere along the beam line, the most efficient way to reject them is to require that the interaction occurs close to the expected bunch-crossing position. The position of the collision along the beam pipe is determined from the vertex position reconstructed correlating SPD tracklets, with a precision of about 0.2 cm. Beam-gas interactions far from the IP are vetoed by the time difference in the V0-A and V0-C detectors. The vertex is required to be within 4 cm of the nominal IP position to reduce the contribution from beam-gas interactions and to remove acceptance gaps in the pseudorapidity coverage of the SPD and FMD, since the acceptance depends on the vertex position.

Even though runs with very low μ (average 0.04) were chosen, a residual background from pileup events remains. The majority of pileup events are identified and removed by searching for additional vertices in the same event. It is required that the uncertainty on the measurement of the longitudinal vertex position is less than 0.2 cm to have the most accurate determination of the vertex. Events with an additional vertex separated by more than 0.8 cm from the main one and containing at least three attached tracklets are tagged as pileup and removed from the analysis. Dedicated simulations show that the probability for the pileup event to pass this selection criteria is at most 10% and the residual pileup does not exceed 10% up to the highest multiplicities kept in this analysis. Therefore, the overall pileup contribution does not exceed 0.2% for μ = 0.04 and, because it is covered by systematic uncertainties for all multiplicities, no correction is applied for this bias.

Unfolding

The FMD had nearly 100% azimuthal acceptance, but the SPD had a significant number of modules excluded from read-out that must be accounted for. On the other hand, interactions in detector material increase the detected number of charged particles, in particular in the FMD. A good understanding of the detector acceptance and of the number of secondary particles which hit the FMD and the SPD is crucial.

The main ingredients necessary to evaluate the primary multiplicity distributions are the raw (detected) multiplicity distributions and a matrix, which maps the measured multiplicity to the number of charged-primary particles distributions, called true. The raw multiplicity distributions are determined by counting the number of clusters in the SPD acceptance, the number of signals passing selection criteria in the FMD, or the average between the two if the acceptance of the SPD and FMD overlaps. The response of the detector is determined by the matrix R mt , which corresponds to the probability that an event with true multiplicity t and measured multiplicity m occurs. This matrix is obtained using PYTHIA ATLAS-CSC flat tune [START_REF] Enterria | Constraints from the first LHC data on hadronic event generators for ultra-high energy cosmic-ray physics[END_REF] simulations in which the generated particles are transported through the experimental setup using the GEANT3 [START_REF] Brun | GEANT Detector Description and Simulation Tool[END_REF] software package. The same reconstruction algorithm is used for simulations of real data. Experimental conditions and detector settings at the time of data-taking at a center-of-mass energy of √ s = 0.9, 7, and 8 TeV are simulated when evaluating the response matrices. Figure 1 shows two different response matrices for different pseudorapidity ranges. The left panel of Fig. 1 shows the response matrix obtained for the |η| < 2.0. In this range, the unfolding increases the multiplicity on average because of the acceptance gaps in the SPD. When the extended pseudorapidity range, |η| < 3.4, is used, the number of detected counts exceeds on average the number of true counts as the secondary particles in the FMD dominate the bias. This is shown in the right panel of Fig. 1.

A method based on Bayes' Theorem [START_REF] Agostini | Improved iterative Bayesian unfolding[END_REF] is used to derive the final multiplicity distributions. Bayes' Theorem states Generated-charged multiplicity Reconstructed-charged multiplicity ALICE Simulation = 0.9 TeV s pp, 1 Response matrices obtained propagating Monte Carlo generated events, in this case with the PYTHIA ATLAS-CSC flat tune [START_REF] Enterria | Constraints from the first LHC data on hadronic event generators for ultra-high energy cosmic-ray physics[END_REF] for the non-single-diffractive event class selection. Left: Matrix includ-ing the overlap region between SPD and FMD. Right: Matrix for the region where the majority of the counts are from the FMD. The diagonal (generated=reconstructed) is plotted as a black dotted line that the conditional probability P(A|B) (probability of A if B is true) can be written as

| < 2.0 η | | < 3.4 η | Fig.
P(A|B) = P(B|A)P(A) P(B) , (1)
in which P(A) and P(B) are the independent probabilities of A and B, and P(B|A) is the probability of B if A is true. A can be identified as a certain true multiplicity, while B is the measured multiplicity. The conditional probability P(B|A) is the response matrix of the detector, and can then be computed. Equation 1 is restated as

R tm = R mt P t t' R mt P t , ( 2
)
where P t is an a priori guess of the true distribution and R tm is the matrix of probabilities that allows one to compute the true multiplicity distribution from the measured one. The unfolded distribution, U t , is then obtained from

U t = m R tm M m , (3)
in which M m is the measured distribution. The obtained U t is used as a priori probability for the next iteration. The number of iterations is fixed to 10. This parameter has been chosen by examining the optimal performance obtained from simulation studies, performing closure tests using different number of iterations.

Event selection efficiency

The probability that an event is triggered depends on the multiplicity of charged particles. At high multiplicities, it is more probable that one of the trigger detectors is fired. At low multiplicities large trigger inefficiencies for finding events exist and must be corrected for. The event selection efficiency, TRIG , is defined via simulations as

TRIG = N ev,reco (TRIG & |v z,reco | < 4 cm) N ev,gen (TRIG & |v z,gen | < 4 cm) , ( 4
)
where the numerator is the number of reconstructed events with the selected hardware trigger condition (MB AND or MB OR ) and with the reconstructed vertex less than 4 cm from the nominal IP, in longitudinal direction. There is a dependence in the z vertex distribution and selecting v z,reco introduces a bias in the efficiency. The effect is visible only for narrow vertex selections, and it is not relevant for |v z | < 4 cm. The denominator is a similar quantity, but for the generated sample (inelastic or non-single-diffractive events). The unfolded distribution is corrected for the vertex and trigger inefficiency by dividing each multiplicity bin by its TRIG value.

The efficiencies used are shown in Fig. 2 for 0.9 and 7 TeV for the range |η| < 3.0. Both the INEL and NSD efficiencies are displayed. The points are obtained by averaging the efficiencies found with the PYTHIA Perugia 0 [START_REF] Skands | Tuning Monte Carlo generators: the perugia tunes[END_REF] and the PHOJET [START_REF] Bopp | Rapidity gaps and the PHOJET Monte Carlo[END_REF] diffraction tuned event generators. Diffraction was accounted for using the Kaidalov-Poghosyan model [START_REF] Kaidalov | Description of soft diffraction in the framework of reggeon calculus: Predictions for LHC[END_REF] to tune the diffractive processes. The event generators are adjusted to reproduce the measured diffraction cross-sections and the shapes of the diffractive masses. The cross-section ratios are σ SD /σ INEL 0.20 for upper diffractive mass limit of M X < 200 GeV/c 2 , and σ DD /σ INEL 0.11 for a pseudorapidity gap of η > 3, as measured at the LHC [START_REF] Abelev | Measurement of inelastic, single-and double-diffraction cross sections in proton-proton collisions at the LHC with ALICE[END_REF].

Systematic uncertainties

The steps involved in the analysis depend on the knowledge of the detector response to charged particles. The uncertain-ties in Table 2 are purely model dependent and related to how diffraction, and soft QCD in general, are processed in the two generators used to determine the efficiency uncertainty. The difference between PYTHIA Perugia 0 and PHOJET diffraction tuned generators, used to determine this uncertainty, is larger for small values of N ch . Therefore, the uncertainty mostly influences the first bins of the multiplicity distributions. Table 2 reports the values for charged-multiplicity of 0, 1, and 2. In general, the Lorentz boost of the diffracted system increases with increasing center-of-mass energies, and single and double diffraction contributions are smaller when going to higher energies. At wider pseudorapidity ranges there are higher chances of including diffractive events in the distribution. We observe that the uncertainty for NSD events at -3.4 < η < + 5.0 is higher for lower energy in one multiplicity bin, where the description of diffraction differs the most among PYTHIA and PHOJET. Systematic effects from different sources related to run conditions could produce biases in the number of detected particles. To investigate such effects, the fluctuations in the results are examined for all three energies by splitting the data set into two separate samples with similar beam conditions, which are then unfolded with two different response matrices. The response matrices are calculated from simulations relative to the conditions of the runs that are used to unfold. The two resulting unfolded distributions are then averaged bin by bin. For √ s = 0.9 and 8 TeV, the run-to-run fluctuations are found to be negligible up to the value of N ch in which statistical uncertainties become large. For √ s = 7 TeV, however, run-to-run fluctuations of around 10-15% in the low N ch bins are found.

As discussed in Sect. 3.2, an accurate detector description is crucial in determining the number of particles created in interactions with detector material in order to retrieve the primary distribution. For the SPD, little material (besides the beam pipe) exists between the detector and the interaction point, whereas a significant amount of material is present between the FMD and the interaction point. An estimate of the amount of material versus η was done using special satellite collisions [START_REF] Adam | Centrality evolution of the charged-particle pseudorapidity density over a broad pseudorapidity range in Pb-Pb collisions at √ s NN = 2.76 TeV[END_REF], which occur away from the nominal IP and thereby reduce the amount of traversed material. The result was that for -3.4 < η < -1.7 the material was underestimated in the simulations of the experiment by a maximum value of 14%. For 1.7 < η < 5.0, the material was estimated with ± 7% precision. It is possible to correct for the measurements of the first moment of the distribution (pseudorapidity density [START_REF] Adam | Centrality evolution of the charged-particle pseudorapidity density over a broad pseudorapidity range in Pb-Pb collisions at √ s NN = 2.76 TeV[END_REF]). On the contrary, higher order effects are non-trivial, making it impossible to directly correct in this case for the amount of material. Instead, the entire raw distribution was unfolded with two response matrices, one which increases the material by 14% and the other which decreases the material by 7%. The difference between the results using the two different matrices determined the maximum systematic error contribution from the material budget.

The last uncertainty was determined by varying the selection criteria used to determine which signals correspond to single particles in the FMD (described in Sect. 2.3) by 5%. This value is the maximum variation of the fit parameters to the energy distributions [START_REF] Abbas | Centrality dependence of the pseudorapidity density distribution for charged particles in Pb-Pb collisions at √ s NN = 2.76 TeV[END_REF] within each of the three data taking periods.

The systematic uncertainties from the three sources described along with the one from the efficiency correction are summed in quadrature (see Table 3). The methods to determine the systematic uncertainties from material budget give the largest possible variations. To convert the variations to a root-mean-square value, they have been divided by √ 3 considering that the variation is flat and was taken from the mean bin-by-bin value as the reference, not the full spread. Three particular multiplicities are considered when reporting the systematic uncertainties in Table 3: the 0-bin, the mean value m , and the value in which the probabil- ity is very low (P(N ch ) = 10 -4 ). For P(N ch ) < 10 -4 the uncertainties grow rapidly and the level of the systematic uncertainty depends strongly on the multiplicity. The P(N ch ) region with lowest uncertainty is near the mean of the distribution.

Results

The multiplicity distributions have been measured for the three event classes (INEL >0, INEL, and NSD) for pp collisions at √ s = 0.9, 7, and 8 TeV. To extract the relative contributions from hard and soft processes, the distributions are fitted with double Negative Binomial Distributions. The results are also compared with the LHC measurements done by CMS and with distributions obtained from models including the IP-Glasma, which is based on the Color Glass Condensate (CGC) [START_REF] Iancu | The Color glass condensate and highenergy scattering in QCD[END_REF]. Fig. 8 Charged-particle multiplicity distributions for pp collisions at √ s = 7 TeV compared to distributions from the IP-Glasma model with the ratio between Q s and the color charge density either fixed (green stars), allowed to fluctuate with a Gaussian (blue squares) [START_REF] Schenke | Multiplicity distributions in p+p, p+A and A+A collisions from Yang-Mills dynamics[END_REF] or with additional fluctuations of proton saturation scale (black diamonds) [START_REF] Mclerran | Intrinsic fluctuations of the proton saturation momentum scale in high multiplicity p+p collisions[END_REF] The colored bands represent the combined systematic and statistical uncertainties. The distributions are scaled by powers of 10 for clarity. The lines show fits to double NBDs, explained in Sect. 5.2. Extending the pseudorapidity coverage with respect to the previous ALICE publications [START_REF] Aamodt | Charged-particle multiplicity measurement in proton-proton collisions at √ s = 0.9 and 2[END_REF][START_REF] Aamodt | Charged-particle multiplicity measurement in proton-proton collisions at √ s = 7 TeV with ALICE at LHC[END_REF][START_REF] Adam | Charged-particle multiplicities in proton-proton collisions at √ s = 0.9 to 8 TeV[END_REF] allows to extend the high-multiplicity reach.

In Fig. 6 (left panel), the comparison to published CMS data [START_REF] Khachatryan | Charged particle multiplicities in pp interactions at √ s = 0.9, 2.36, and 7 TeV[END_REF] is shown for the NSD event class at √ s = 0.9 TeV in three η ranges. Discrepancies are observed in the comparison with CMS, both in the first bins and, especially, in the tails. Differences in the first bins can be attributed presumably to the different models used to describe the diffraction masses. In particular, CMS is not using any diffraction tuned simulation. Moreover, in this paper, single-diffractive events include a cut on diffractive mass [START_REF] Kaidalov | Description of soft diffraction in the framework of reggeon calculus: Predictions for LHC[END_REF], which is different from what CMS has used. In the tails, the CMS data are systematically lower, due to normalization. This behavior was also observed when comparing CMS distributions to those obtained in a narrower pseudorapidity range where only SPD tracklets are used [START_REF] Adam | Charged-particle multiplicities in proton-proton collisions at √ s = 0.9 to 8 TeV[END_REF] as shown in the distribution for |η| < 1.0 in Fig. 6.

In the right panel of Fig. 6, the comparison to CMS at 7 TeV [START_REF] Khachatryan | Charged particle multiplicities in pp interactions at √ s = 0.9, 2.36, and 7 TeV[END_REF] is shown. Good agreement with CMS is observed except in the very first bins presumably due to the different treatment of diffraction masses. The measurement reported in this manuscript, performed with the SPD clusters, agrees with the analysis performed on SPD tracklets [START_REF] Adam | Charged-particle multiplicities in proton-proton collisions at √ s = 0.9 to 8 TeV[END_REF] within systematic uncertainties.

In Fig. 7, comparisons with distributions obtained with the PYTHIA 6 Perugia 0 tune [START_REF] Skands | Tuning Monte Carlo generators: the perugia tunes[END_REF], PHOJET [START_REF] Bopp | Rapidity gaps and the PHOJET Monte Carlo[END_REF], PYTHIA 8 Monash tune [START_REF] Sjstrand | An introduction to PYTHIA 8.2[END_REF] and EPOS LHC [START_REF] Pierog | EPOS LHC: test of collective hadronization with data measured at the CERN large hadron collider[END_REF] models are shown for INEL events at 0.9 TeV (left plot) and 7 TeV (right plot). At 0.9 TeV, PHOJET, PYTHIA 8 Monash tune and EPOS LHC cannot reproduce the tails, and the lowest values of the multiplicity distributions, while PYTHIA 6 Perugia 0 tune does not reproduce the data at all. At 7 TeV, both PHOJET and PYTHIA 6 strongly underestimate the tails of the multiplicity distributions. PYTHIA 8, with the Monash tune that uses LHC data, reproduces the tails for the wider pseudorapidity range, but shows an enhancement in the peak region. EPOS LHC models the distributions well, both in the first bins, which are dominated by diffractive events, and in the tails.

The evolution of the multiplicity distributions with the center-of-mass energy √ s can be studied using the KNO variable N ch / N ch [START_REF] Koba | Scaling of multiplicity distributions in high-energy hadron collisions[END_REF]. KNO scaling violation is observed if the tails of the distributions increase with increasing energy. The violation increases when going to larger pseudorapidity ranges. This behavior was already observed at central rapidities [START_REF] Adam | Charged-particle multiplicities in proton-proton collisions at √ s = 0.9 to 8 TeV[END_REF], and, therefore, it is not investigated any further.

The multiplicity distributions at 7 TeV are compared to those from the IP-Glasma model [START_REF] Schenke | Multiplicity distributions in p+p, p+A and A+A collisions from Yang-Mills dynamics[END_REF]. This model is based on the Color Glass Condensate (CGC) [START_REF] Iancu | The Color glass condensate and highenergy scattering in QCD[END_REF]. It has been shown that particle multiplicities are generated following an NBD within the CGC framework [START_REF] Gelis | Glittering glasmas[END_REF]. Moreover, the multiplicity distribution generated by the decay of the Glasma flux tubes [START_REF] Mclerran | A Brief Introduction to the Color Glass Condensate and the Glasma[END_REF] is a NBD with parameter k (see following Section) ∝ Q 2 s S ⊥ , in which Q s is the gluon saturation scale and S ⊥ is the transverse overlap area of the collision [START_REF] Schenke | Multiplicity distributions in p+p, p+A and A+A collisions from Yang-Mills dynamics[END_REF]. The CGC based IP-Glasma model, therefore, has a built-in source of multiplicity fluctuations. In Fig. 8, the distribution for |η| < 2.0 is shown together with the IP-Glasma model distributions as a function of the KNO variable N ch / N ch . The IP-Glasma distribution, shown in green stars, generated with a fixed ratio between Q s and density of color charge, thus introducing no fluctuations. The blue squares distribution is generated with fluctuations of the color charge density around the mean value following a Gaussian distribution 123 with a width of σ = 0.09. The black diamonds distribution includes an additional source of fluctuations, dominantly of non-perturbative origin, from stochastic splitting of color dipoles that is not accounted for in the conventional frameworks of CGC [START_REF] Mclerran | Intrinsic fluctuations of the proton saturation momentum scale in high multiplicity p+p collisions[END_REF]. In the IP-Glasma model shown, the evolution of color charges in the rapidity direction still needs to be implemented and in the present model the low multiplicity bins are not reproduced for the wide pseudorapidity range presented.

Double NBD fits

Already at 0.9 TeV for |η| < 1.3 in NSD events, ALICE observed that the distributions are not well described by a single NBD parameterization [START_REF] Aamodt | Charged-particle multiplicity measurement in proton-proton collisions at √ s = 0.9 and 2[END_REF]. For this reason, a parameterization with the sum of two NBDs has been performed for the NSD, INEL and INEL > 0 event samples. The fits are plotted together with the results in Figs. 3,4, 5, and parameters are given in Table 4 for the NSD event sample, in Table 5 for the INEL event sample and in Table 6 for the INEL >0 event sample. The distributions have been fitted using the function

P(n) = λ [αP NBD (n, n 1 , k 1 )+(1 -α)P NBD (n, n 2 , k 2 )] , (5)
where

P NBD (n, n , k) = (n + k) (k) (n + 1) n k + n n k k + n k . ( 6
)
NBDs do not describe the bin with N ch = 0 and the first bins for the wider pseudorapidities, in which there is a rise in the number of events measured due to diffractive events (bins removed from the fit). To account for this a normalization factor λ is introduced [START_REF] Adam | Charged-particle multiplicities in proton-proton collisions at √ s = 0.9 to 8 TeV[END_REF]. The systematic uncertainty for the efficiency correction produces a fully correlated shift of the distribution, and therefore is omitted in the fitted data sample. The other sources of uncertainty are kept in the data. From fitting with the above uncertainties included, one obtains the parameters in bold. The parameters of two extreme cases are printed as well in Tables 4, 5 and 6 to allow the reader to have an estimate on how much the fit parameters change due to the remaining correlations present. The upper and lower case correspond to fitting the distribution obtained adding and subtracting material in the detector. The behavior of the fit parameters is consistent with what was observed by CMS [START_REF] Khachatryan | Charged particle multiplicities in pp interactions at √ s = 0.9, 2.36, and 7 TeV[END_REF][START_REF] Ghosh | Negative binomial multiplicity distribution in protonproton collisions in limited pseudorapidity intervals at LHC up to √ s = 7 TeV and the clan model[END_REF]. The average multiplicity of the soft (first) component, n 1 , increases with increasing energy and pseudorapidity range. The parameter k 1 increases with increasing pseudorapidity range and decreases with centerof-mass energy. For the semi-hard (second) component, the n 2 parameter behaves similarly to n 1 . Moreover, it is noted that n 2 3 n 1 . The parameter α reflects the fraction of the soft events. The percentage of soft events decreases with increasing pseudorapidity range and energy indicating that a higher percentage of semi-hard events is then present.

For KNO scaling to hold, the parameter k must be constant with energy. This means that KNO scaling is violated for all pseudorapidity ranges for both the soft and semi-hard components in the NSD, INEL and INEL >0 event samples. Three possible scenarios are proposed in the model by Ugoccioni and Giovannini [START_REF] Giovannini | Possible scenarios for soft and semihard components structure in central hadron hadron collisions in the TeV region[END_REF] concerning KNO scaling in the semihard component. Parameter k 2 can be constant, decrease linearly with increasing energy showing violation, or decrease with energy asymptotically to a constant value showing weak violation. The last scenario appears to be in agreement with the values obtained for k 2 , since the decrease from 0.9 to 7 TeV is very strong, while the values are compatible for 7 and 8 TeV. The analysis of 13 TeV data will reveal if the values will be compatible also in that case, or if it is only due to the vicinity of the 7 and 8 TeV energies.

Summary and conclusions

Data from the SPD and the FMD were used to access a uniquely wide pseudorapidity coverage at the LHC of more than eight units in pseudorapidity, from -3.4 < η < 5.0. The charged-particle multiplicity distributions obtained from these data were presented for three pp collision energies, √ s = 0.9, 7, and 8 TeV, and for three different event classes, INEL, INEL > 0, and NSD. The results shown extend the pseudorapidity coverage of the earlier results published by ALICE and CMS, and, consequently, the high-multiplicity reach. The extension of pseudorapidity coverage has higher systematic uncertainties due to the unknown fraction of the material budget in front of the FMD, estimated to be up to 14%.

The multiplicity distributions for 7 TeV collisions measured for NSD events are in agreement with those from CMS. For 0.9 TeV, the results shown are systematically lower in the low-multiplicity bins when compared to the CMS measurement. This is most probably due to a different estimation of the diffractive masses distribution used to tune the simulations for the efficiency correction between CMS and ALICE. At 0.9 TeV, PYTHIA, PHOJET and EPOS LHC cannot reproduce the multiplicity distributions, although PHOJET is closer to the results being tuned using LHC data. At 7 TeV, PYTHIA 6 and PHOJET strongly underestimate the fraction of high multiplicity events. PYTHIA 8 slightly underestimates the tails of the distributions, while EPOS LHC reproduces both the low and the high multiplicity events, showing better capabilities to model diffraction. The Color Glass Condensate based IP-Glasma model produces distribu-123 tions, which underestimate the fraction of high multiplicity events, but introducing fluctuations in the saturation momentum allows to reproduce the measurements better. Double Negative Binomial distributions composed of a soft and hard component are fitted to the measured distributions. The fraction of soft events decreases with increasing pseudorapidity range and with the increasing collision energy. KNO scaling is violated for the three considered event classes, at all the collision energies probed. The 13 TeV data analysis will help delving into the description of the components of the multiplicity distributions.

|Fig. 2

 2 Fig.2Event selection efficiencies for 0.9 and 7 TeV for both INEL and NSD event samples as a function of the number of primary-charged particles for the |η| < 3.0 range

Fig. 3 Fig. 4

 34 Fig. 3 Charged-particle multiplicity distributions for NSD (top left), INEL (top right) and INEL >0 (bottom) pp collisions at √ s = 0.9 TeV. The lines show fits to double NBDs. Ratios of the data to the fits are also

Fig. 5

 5 Fig. 5 Charged-particle multiplicity distributions for NSD (top left), INEL (top right) and INEL >0 (bottom) pp collisions at √ s = 8 TeV. The lines show fits to double NBDs. Ratios of the data to the fits are also

5. 1

 1 Multiplicity distributions In Figs. 3, 4 and 5, the obtained multiplicity distributions for 0.9, 7, and 8 TeV with NSD, INEL and INEL >0 triggers are shown for five pseudorapidity ranges, |η| < 2.0, |η| < 2.4, |η| < 3.0, |η| < 3.4 and -3.4 < η < + 5.0.

Fig. 6 Fig. 7

 67 Fig.[START_REF] Aad | Charged-particle multiplicities in pp interactions measured with the ATLAS detector at the LHC[END_REF] Comparison of the multiplicity distributions for NSD pp collisions at √ s = 0.9 TeV (left) and 7 TeV (right) with CMS[START_REF] Khachatryan | Charged particle multiplicities in pp interactions at √ s = 0.9, 2.36, and 7 TeV[END_REF] measurements in the same pseudorapidity ranges and previous ALICE measurements[START_REF] Adam | Charged-particle multiplicities in proton-proton collisions at √ s = 0.9 to 8 TeV[END_REF]. Combined systematic and statistical uncertainties are shown as bands ) ch N P(

Table 1

 1 Data samples used in this analysis. For each center-of-mass energy, the total number of selected minimum-bias (MB) events along with the average number of interactions per bunch crossing, μ , are

	listed √ s (TeV)	Selected MB events	μ
	0.9	7.4 × 10 6	0.04 ± 0.01
	7	6 1 × 10 6	0.04 ± 0.01
	8	2 6 × 10 6	0.02 ± 0.01
	central-diffractive. In Regge theory

Table 2

 2

	Systematic uncertainties (in percent) for the	η range	Event class	√	s (TeV)
	efficiency correction, for the			0.9			7			8
	INEL, NSD and INEL> 0 event classes. Numbers are given for multiplicity 0, 1, and 2			N ch 0	1	2	0	1	2	0	1	2
		Efficiency uncertainty
		|η| < 2.0	INEL	15.3	6.2	3.0	27.0 11.5 5.0	28.0 12.7 8.7
			NSD	5.3	2.7	0.5	9.8	6.9	0.5	9.8	8.8	7.1
			INEL > 0	-		6.2	3.1	-	11.5 5.0	-	12.7 8.7
		|η| < 2.4	INEL	19.9	14.1 8.5	26.9 20.6 12.0 28.1 22.5 16.6
			NSD	8.1	10.7 5.7	14.4 17.6 8.5	14.6 19.8 14.2
			INEL > 0	-		14.1 8.5	-	20.6 12.0 -	22.5 16.6
		|η| < 3.0	INEL	25.8	24.4 20.7 30.4 33.5 26.0 32.1 35.2 28.9
			NSD	15.4	22.2 18.4 19.3 29.1 25.1 18.8 31.0 28.1
			INEL > 0	-		24.4 20.7 -	33.5 26.0 -	35.2 28.9
		|η| < 3.4	INEL	31.2	34.5 29.6 50.4 40.3 35.4 53.4 41.4 37.2
			NSD	17.3	35.1 27.7 22.4 39.0 32.0 21.0 40.2 33.9
			INEL > 0	-		34.4 29.6 -	40.3 35.4 -	41.4 37.2
		-3.4 < η < + 5.0 INEL	48.3	45.1 36.7 71.1 43.1 44.3 75.4 45.7 47.7
			NSD	35.3	64.9 34.8 55.5 31.4 39.8 50.9 34.9 43.5
			INEL > 0	-		45.1 36.7 -	43.1 44.3 -	45.7 47.7
	between the two event generators and are only relevant at low
	multiplicity. The efficiency of NSD trigger requiring signal
	in V0-A and V0-C detectors, on both sides of the IP, is lower
	at low multiplicities than that of INEL trigger, which requires
	response of at least one V0. For N ch 20 at the widest pseu-
	dorapidity ranges probed, both efficiencies reach 100% and
	the corresponding systematic uncertainty becomes negligi-
	ble.

Table 3

 3

	Total systematic uncertainties (in percent), for	η range	Event class	√	s (TeV)
	the INEL, NSD and INEL> 0			0.9			7	8
	event classes. Numbers are given at multiplicity values of 0, the mean m , and when			0		m	N ch 10 -4 0	N ch P(N ch ) m 10 -4	0	m	10 -4
	P(N ch ) = 10 -4	Total uncertainty
		|η| < 2.0	INEL	16.6	2.0 12.2	27.4 2.5 28.8	27.1 2.9 17.6
			NSD	7.6	1.3 12.3	11.0 4.4 30.2	9.8	4.6 19.0
			INEL >0	-		1.9 12.9	-	4.0 30.2	-	4.2 17.9
		|η| < 2.4	INEL	20.8	2.3 21.2	27.3 2.9 37.7	28.1 2.2 34.6
			NSD	10.0	3.1 18.0	15.1 4.4 39.2	14.6 7.4 36.9
			INEL >0	-		2.7 22.2	-	4.6 40.1	-	7.4 35.8
		|η| < 3.0	INEL	26.2	1.8 26.1	30.8 4.9 43.9	32.1 4.8 43.6
			NSD	16.1	2.6 26.8	19.9 5.6 44.2	18.8 7.0 43.8
			INEL >0	-		2.7 26.9	-	5.9 52.8	-	6.7 45.0
		|η| < 3.4	INEL	31.5	2.2 27.3	50.8 6.1 51.9	53.4 5.3 50.1
			NSD	17.8	2.4 28.6	23.1 6.1 53.5	21.1 6.1 52.9
			INEL >0	-		2.4 28.7	-	5.9 52.8	-	5.9 51.5
		-3.4 < η < + 5.0 INEL	48.4	1.3 29.8	71.5 5.4 60.5	75.4 3.9 57.4
			NSD	35.6	2.1 30.7	56.0 6.9 63.8	50.9 7.0 61.5
			INEL >0	-		2.4 30.8	-	5.7 63.4	-	5.7 59.9

Table 4

 4 Double NBD fit parameters for multiplicity distributions, NSD events. The central values of the parameters are printed in bold with their relative uncertainty, while the top row shows the upper bound fit parameters, and the bottom row the lower bound ones

	√ s (TeV)	η range	λ	α	n 1	k 1	n 2	k 2
	0.9	|η| < 2.0	0.95	0.49	10.16	3.60	24.86	3.83
			0.94 ± 0.01	0.52 ± 0.05	10.28 ± 0.69	3.49 ± 0.30	25.08 ± 1.96	3.96 ± 0.63
			0.94	0.55	10.42	3.38	25.37	4.13
		|η| < 2.4	0.95	0.38	11.76	4.12	28.00	3.27
			0.95 ± 0.01	0.27 ± 0.12	10.94 ± 1.14	4.76 ± 2.91	25.35 ± 4.40	2.80 ± 0.96
			0.95	0.09	10.75	25.98	21.79	2.20
		|η| < 3.0	0.94	0.27	14.41	5.85	33.00	2.99
			0.94 ± 0.01	0.17 ± 0.06	13.68 ± 0.32	9.81 ± 9.82	29.62 ± 2.85	2.54 ± 0.51
			0.95	0.11	13.48	21.75	27.58	2.35
		|η| < 3.4	0.94	0.37	17.09	4.74	38.94	3.43
			0.94 ± 0.01	0.26 ± 0.09	15.80 ± 0.95	5.95 ± 2.96	35.26 ± 4.35	2.98 ± 0.77
			0.94	0.20	14.99	18.98	32.83	2.78
		-3.4 < η < + 5.0	0.95	0.42	20.31	5.17	46.55	4.21
			0.95 ± 0.01	0.36 ± 0.07	19.27 ± 1.45	5.48 ± 1.18	43.41 ± 4.93	3.85 ± 1.03
			0.94	0.16	18.09	15.32	36.64	2.91
	7	|η| < 2.0	0.96	0.36	10.72	2.92	37.66	2.53
			0.94 ± 0.01	0.37 ± 0.02	10.63 ± 0.43	2.91 ± 0.22	36.84 ± 1.41	2.51 ± 0.20
			0.93	0.38	10.56	2.89	36.05	2.51
		|η| < 2.4	0.97	0.36	13.19	3.13	45.86	2.61
			0.95 ± 0.01	0.35 ± 0.03	12.73 ± 0.68	3.17 ± 0.40	43.05 ± 2.21	2.47 ± 0.27
			0.92	0.34	12.25	3.25	40.12	2.36
		|η| < 3.0	0.97	0.36	16.73	3.22	57.56	2.75
			0.94 ± 0.01	0.32 ± 0.03	15.55 ± 0.94	3.52 ± 0.58	52.08 ± 2.93	2.45 ± 0.29
			0.91	0.26	14.28	4.36	45.78	2.15
		|η| < 3.4	0.97	0.38	19.45	3.15	65.72	2.89
			0.94 ± 0.01	0.31 ± 0.03	17.43 ± 1.05	3.68 ± 0.65	57.38 ± 3.33	2.43 ± 0.30
			0.91	0.21	15.48	5.83	48.46	2.02
		-3.4 < η < + 5.0	0.98	0.37	23.17	3.52	77.02	3.11
			0.94 ± 0

.01 0.30 ± 0.03 20.74 ± 1.28 4.18 ± 0.77 66.40 ± 4.08 2.54 ± 0.34

			0.90	0.16	18.52	9.63	53.51	1.97
	8	|η| < 2.0	0.96	0.44	12.58	2.38	42.18	2.97
			0.94 ± 0

.01 0.45 ± 0.03 12.37 ± 0.79 2.38 ± 0.15 41.16 ± 2.01 2.93 ± 0.29

		0.93	0.46	12.21	2.38	40.20	2.91
	|η| < 2.4	0.96	0.37	14.18	2.77	48.59	2.70
		0.94 ± 0.01	0.37 ± 0.04	13.71 ± 1.10	2.75 ± 0.34	45.73 ± 3.15	2.56 ± 0.37
		0.91	0.36	13.20	2.76	42.69	2.47
	|η| < 3.0	0.97	0.31	16.77	3.22	57.63	2.49
		0.94 ± 0.01	0.26 ± 0.03	15.50 ± 0.99	3.63 ± 0.78	51.58 ± 3.25	2.19 ± 0.29
		0.90	0.19	14.20	5.01	44.90	2.92
	|η| < 3.4	0.97	0.33	19.54	3.21	66.21	2.64
		0.93 ± 0.01	0.26 ± 0.03	17.57 ± 0.91	3.84 ± 0.75	57.55 ± 3.17	2.22 ± 0.27
		0.90	0.18	15.93	5.73	49.51	1.95
	-3.4 < η < + 5.0	0.99	0.26	21.79	4.33	73.60	2.47
		0.94 ± 0.01	0.17 ± 0.03	20.11 ± 0.56	6.65 ± 2.45	62.60 ± 3.03	2.00 ± 0.21
		0.90	0.14	19.22	9.96	55.01	1.92

Table 5

 5 Double NBD fit parameters for multiplicity distributions, INEL events. The central values of the parameters are printed in bold with their relative uncertainty, while the top row shows the upper bound fit parameters, and the bottom row the lower bound ones

	√ s (TeV)	η range	λ	α	n 1	k 1	n 2	k 2
	0.9	|η| < 2.0	0.81	0.61	10.25	2.77	26.59	4.24
			0.81 ± 0.01	0.64 ± 0.05	10.44 ± 0.97	2.69 ± 0.20	27.03 ± 2.71	4.45 ± 0.91
			0.81	0.68	10.67	2.61	27.62	4.73
		|η| < 2.4	0.80	0.39	10.72	3.75	27.34	3.07
			0.81 ± 0.01	0.33 ± 0.05	10.12 ± 0.61	4.00 ± 0.93	25.48 ± 2.11	2.80 ± 0.47
			0.81	0.26	9.51	4.72	23.52	2.52
		|η| < 3.0	0.80	0.38	13.81	4.14	34.33	3.20
			0.80 ± 0.01	0.31 ± 0.05	12.91 ± 0.66	4.83 ± 1.39	31.45 ± 2.67	2.83 ± 0.50
			0.80	0.24	12.23	6.29	28.96	2.56
		|η| < 3.4	0.80	0.46	16.52	3.73	40.48	3.67
			0.80 ± 0.01	0.40 ± 0.05	15.30 ± 0.95	4.10 ± 0.63	37.39 ± 2.97	3.33 ± 0.56
			0.80	0.36	14.46	4.54	35.52	3.24
		-3.4 < η < + 5.0	0.80	0.50	19.87	4.20	48.17	4.49
			0.80 ± 0.01	0.46 ± 0.04	18.83 ± 1.08	4.32 ± 0.45	45.16 ± 3.18	4.16 ± 0.69
			0.80	0.43	17.86	4.53	42.56	3.97
	7	|η| < 2.0	0.80	0.42	10.51	2.44	38.31	2.64
			0.79 ± 0.01	0.43 ± 0.02	10.42 ± 0.49	2.43 ± 0.17	37.54 ± 1.56	2.63 ± 0.23
			0.78	0.44	10.35	2.41	36.81	2.64
		|η| < 2.4	0.80	0.39	12.61	2.77	45.69	2.63
			0.79 ± 0.01	0.38 ± 0.03	12.12 ± 0.71	2.79 ± 0.38	42.87 ± 2.36	2.48 ± 0.28
			0.77	0.37	11.62	2.85	39.92	2.37
		|η| < 3.0	0.80	0.40	16.17	2.77	57.70	2.82
			0.78 ± 0.01	0.35 ± 0.03	14.75 ± 1.01	3.13 ± 0.63	51.55 ± 3.28	2.45 ± 0.32
			0.75	0.28	13.38	4.47	44.74	2.11
		|η| < 3.4	0.80	0.35	19.04	2.67	66.42	3.03
			0.78 ± 0.01	0.35 ± 0.03	16.69 ± 1.14	3.15 ± 0.57	57.31 ± 3.63	2.48 ± 0.33
			0.75	0.26	14.82	4.47	48.93	2.11
		-3.4 < η < + 5.0	0.81	0.41	22.34	3.03	77.02	3.20
			0.78 ± 0.01	0.33 ± 0.03	19.54 ± 1.21	3.75 ± 0.77	65.47 ± 4.01	2.53 ± 0.33
			0.75	0.24	17.57	5.70	54.79	2.12
	8	|η| < 2.0	0.81	0.51	11.13	1.89	41.16	3.04
			0.80 ± 0.01	0.52 ± 0.03	10.87 ± 0.84	1.93 ± 0.25	40.01 ± 2.39	2.98 ± 0.35
			0.79	0.52	10.67	1.96	38.95	2.94
		|η| < 2.4	0.80	0.43	12.58	2.38	47.19	2.73
			0.79 ± 0.01	0.42 ± 0.04	12.08 ± 0.98	2.37 ± 0.38	44.39 ± 3.12	2.58 ± 0.37
			0.77	0.43	11.54	2.39	41.38	2.47
		|η| < 3.0	0.80	0.37	15.25	2.72	56.72	2.57
			0.78 ± 0.01	0.32 ± 0.03	13.94 ± 0.85	3.10 ± 0.64	50.50 ± 3.16	2.23 ± 0.29
			0.75	0.26	12.78	4.06	43.94	1.94
		|η| < 3.4	0.80	0.41	18.01	2.65	65.78	2.78
			0.78 ± 0.01	0.34 ± 0.03	16.03 ± 0.86	3.08 ± 0.49	57.14 ± 3.15	2.32 ± 0.28
			0.75	0.27	14.56	3.98	49.55	2.05
		-3.4 < η < + 5.0	0.82	0.37	20.95	3.16	75.52	2.78
			0.78 ± 0.01	0.30 ± 0.03	19.03 ± 0.86	3.78 ± 0.74	65.08 ± 3.74	2.29 ± 0.29
			0.75	0.23	17.61	5.20	55.11	1.99

Table 6

 6 Double NBD fit parameters for multiplicity distributions, INEL >0 events. The central values of the parameters are printed in bold with their relative uncertainty, while the top row shows the upper bound fit parameters, and the bottom row the lower bound ones

	√ s (TeV)	η range	λ	α	n 1	k 1	n 2	k 2
	0.9	|η| < 2.0	1.00	0.53	10.43	3.40	25.55	3.94
			1.00 ± 0.01	0.56 ± 0.04	10.54 ± 0.69	3.31 ± 0.21	25.78 ± 2.04	4.08 ± 0.65
			1.00	0.59	10.67	3.23	26.00	4.26
		|η| < 2.4	1.00	0.49	12.35	3.40	30.27	3.75
			1.00 ± 0.01	0.45 ± 0.06	11.84 ± 1.11	3.37 ± 0.37	28.64 ± 2.93	3.49 ± 0.73
			0.99	0.11	10.56	19.63	22.06	2.23
		|η| < 3.0	0.97	0.32	14.35	4.81	33.95	3.13
			0.97 ± 0.01	0.23 ± 0.07	13.42 ± 0.56	6.30 ± 3.33	30.77 ± 3.00	2.71 ± 0.55
			0.97	0.14	12.99	12.54	28.07	2.41
		|η| < 3.4	0.95	0.40	16.98	4.26	39.78	3.55
			0.95 ± 0.01	0.33 ± 0.05	15.76 ± 0.90	4.80 ± 0.98	36.74 ± 2.96	3.22 ± 0.55
			0.95	0.29	14.88	5.50	34.84	3.11
		-3.4 < η < +5.0	0.93	0.46	20.45	4.61	47.77	4.41
			0.92 ± 0.01	0.41 ± 0.05	19.37 ± 1.10	4.77 ± 0.54	44.72 ± 3.20	4.07 ± 0.68
			0.92	0.37	18.35	5.06	42.06	3.87
	7	|η| < 2.0	1.02	0.37	10.96	3.04	37.57	2.55
			1.00 ± 0.01	0.37 ± 0.02	10.88 ± 0.42	3.03 ± 0.22	36.77 ± 1.41	2.53 ± 0.21
			0.99	0.38	10.81	3.01	36.01	2.53
		|η| < 2.4	1.02	0.36	13.31	3.18	45.58	2.62
			1.00 ± 0.01	0.35 ± 0.03	12.87 ± 0.69	3.21 ± 0.41	42.83 ± 2.27	2.48 ± 0.27
			0.97	0.34	12.41	3.28	39.97	2.38
		|η| < 3.0	1.01	0.36	16.88	3.19	57.27	2.78
			0.98 ± 0.01	0.32 ± 0.03	15.58 ± 1.00	3.56 ± 0.67	51.50 ± 3.09	2.44 ± 0.30
			0.94	0.24	14.14	4.81	44.68	2.10
		|η| < 3.4	0.99	0.39	19.79	3.06	65.81	2.97
			0.96 ± 0.01	0.31 ± 0.03	17.53 ± 1.15	3.63 ± 0.69	57.02 ± 3.52	2.45 ± 0.32
			0.93	0.20	15.45	6.03	48.00	2.01
		-3.4 < η < +5.0	0.98	0.39	23.40	3.31	77.14	3.21
			0.94 ± 0

.01 0.31 ± 0.03 20.70 ± 1.37 3.94 ± 0.72 66.27 ± 4.07 2.60 ± 0.34

			0.90	0.17	18.24	8.39	53.51	2.02
	8	|η| < 2.0	1.01	0.43	10.99	2.54	39.09	2.77
			1

.00 ± 0.01 0.43 ± 0.02 10.83 ± 0.52 2.54 ± 0.18 38.14 ± 1.48 2.73 ± 0.21

		0.99	0.44	10.70	2.55	37.24	2.71
	|η| < 2.4	1.01	0.38	12.92	2.91	46.16	2.61
		0.99 ± 0.01	0.37 ± 0.03	12.45 ± 0.81	2.93 ± 0.39	43.33 ± 2.62	2.47 ± 0.31
		0.96	0.35	11.94	2.99	40.32	2.36
	|η| < 3.0	1.00	0.33	15.78	3.24	55.92	2.50
		0.97 ± 0.01	0.28 ± 0.03	14.61 ± 0.77	3.73 ± 0.80	50.01 ± 2.90	2.19 ± 0.27
		0.94	0.22	13.55	5.02	43.78	1.93
	|η| < 3.4	0.99	0.37	18.60	3.08	64.94	2.71
		0.95 ± 0.01	0.30 ± 0.03	16.79 ± 0.80	3.61 ± 0.60	56.72 ± 2.97	2.28 ± 0.26
		0.92	0.23	15.34	4.83	49.25	2.03
	-3.4 < η < +5.0	0.98	0.34	21.85	3.47	75.56	2.79
		0.93 ± 0.01	0.28 ± 0.03	20.00 ± 0.92	4.08 ± 0.77	65.51 ± 3.77	2.32 ± 0.30
		0.89	0.19	18.43	6.23	54.67	1.96

-10 3 -10 2 -

-10 4 -10 3 -10 2 -

Acknowledgements

The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation (ANSL), State Committee of Science and World Federation of Scientists (WFS), Armenia; Austrian Academy of Sciences and Nationalstiftung für Forschung, Technologie und Entwicklung, Austria; Ministry of Communications and High Technologies, National Nuclear Research Center, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Universidade Federal do Rio Grande do Sul (UFRGS),

Financiadora de Estudos e Projetos (Finep) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Brazil; Ministry of Science & Technology of China (MSTC), National Natural Science Foundation of China (NSFC) and Ministry of Education of China (MOEC) , China; Ministry of Science, Education and Sport and Croatian Science Foundation, Croatia; Ministry of Education, Youth and Sports of the Czech Republic, Czech Republic; The Danish Council for Independent Research-Natural Sciences, the Carlsberg Foundation and Danish National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Finland; Commissariat à l'Energie Atomique (CEA) and Institut National de Physique Nucléaire et de Physique des Particules (IN2P3) and Centre National de la Recherche Scientifique (CNRS), France; Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (BMBF) and GSI Helmholtzzentrum für Schwerionenforschung GmbH, Germany; General Secretariat for Research and Technology, Ministry of Education, Research and Religions, Greece; National Research, Development and Innovation Office, Hungary; Department of Atomic Energy Government of India (DAE) and Council of Scientific and Industrial Research (CSIR), New Delhi, India; Indonesian Institute of Science, Indonesia; Centro Fermi-Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi and Istituto Nazionale di Fisica Nucleare (INFN), Italy; Institute for Innovative Science and Technology , Nagasaki Institute of Applied Science (IIST), Japan Society for the Promotion of Science (JSPS) KAK-ENHI and Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan; Consejo Nacional de Ciencia (CONA-CYT) y Tecnología, through Fondo de Cooperación Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de Asuntos del Personal Academico (DGAPA), Mexico; Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council of Norway, Norway; Commission on Science and Technology for Sustainable Development in the South (COMSATS), Pakistan; Pontificia Universidad Católica del Perú, Peru; Ministry of Science and Higher Education and National Science Centre, Poland; Korea Institute of Science and Technology Information and National Research Foundation of Korea (NRF), Republic of Korea; Ministry of Education and Scientific Research, Institute of Atomic Physics and Romanian National Agency for Science, Technology and Innovation, Romania; Joint Institute for Nuclear Research (JINR), Ministry of Education and Science of the Russian Federation and National Research Centre Kurchatov Institute, Russia; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; National Research Foundation of South Africa, South Africa; Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba, Ministerio de Ciencia e Innovacion and Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Spain; Swedish Research Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European Organization for Nuclear Research, Switzerland; National Science and Technology Development Agency (NSDTA), Suranaree University of Technology (SUT) and Office of the Higher Education Commission under NRU project of Thailand, Thailand; Turkish Atomic Energy Agency (TAEK), Turkey; National Academy of Sciences of Ukraine, Ukraine; Science and Technology Facilities Council (STFC), United Kingdom; National Science Foundation of the United States of America (NSF) and United States Department of Energy, Office of Nuclear Physics (DOE NP), United States of America.

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