Forward-backward multiplicity correlations in pp collisions at $\sqrt{s}$=0.9, 2.76 and 7 TeV

The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at $\sqrt{s}=0.9$, 2.76 and 7 TeV. The measurement is performed in the central pseudorapidity region ($|\eta|<0.8$) for the transverse momentum $p_{\rm T}>0.3$ GeV/$c$. Two separate pseudorapidity windows of width ($\delta \eta$) ranging from 0.2 to 0.8 are chosen symmetrically around $\eta=0$. The multiplicity correlation strength ($b_{\rm cor}$) is studied as a function of the pseudorapidity gap ($\eta_{\rm gap}$) between the two windows as well as the width of these windows. The correlation strength is found to decrease with increasing $\eta_{\rm gap}$ and shows a non-linear increase with $\delta\eta$. A sizable increase of the correlation strength with the collision energy, which cannot be explained exclusively by the increase of the mean multiplicity inside the windows, is observed. The correlation coefficient is also measured for multiplicities in different configurations of two azimuthal sectors selected within the symmetric FB $\eta$-windows. Two different contributions, the short-range (SR) and the long-range (LR), are observed. The energy dependence of $b_{\rm cor}$ is found to be weak for the SR component while it is strong for the LR component. Moreover, the correlation coefficient is studied for particles belonging to various transverse momentum intervals chosen to have the same mean multiplicity. Both SR and LR contributions to $b_{\rm cor}$ are found to increase with $p_{\rm T}$ in this case. Results are compared to PYTHIA and PHOJET event generators and to a string-based phenomenological model. The observed dependencies of $b_{\rm cor}$ add new constraints on phenomenological models.


Introduction
We report a detailed study of correlations between multiplicities in pp collisions at 0.9, 2.76 and 7 TeV. The correlations are obtained from event-by-event multiplicity measurements in pseudorapidity (η) and azimuth (ϕ) separated intervals. The intervals are selected one in the forward and another in the backward hemispheres in the center-of-mass system, therefore the correlations are referred to as forward-backward (FB) correlations.
The FB correlation strength is characterized by the correlation coefficient, b corr , which is obtained from a linear regression analysis of the average multiplicity measured in the backward rapidity hemisphere ( n B n F ) as a function of the event multiplicity in the forward hemisphere (n F ): This linear relation (1) has been observed experimentally [1][2][3][4] and is discussed in [5][6][7]. Under the assumption of linear correlation between n F and n B , the Pearson correlation coefficient can be used for the experimental determination of b corr [2]. Since the parameter a is given by a = n B − b corr n F , it adds no additional information and usually is not considered [5][6][7].
Heretofore, FB multiplicity correlations were studied experimentally in a large number of collision systems including e + e − , µ + p, pp, pp and A-A interactions [3,4,[8][9][10][11][12][13]. No FB multiplicity correlations were observed in e + e − annihilation at √ s = 29 GeV. This was interpreted as the consequence of independent fragmentation of the forward and backward jets produced in this process [14]. In contrast, in pp collisions at the ISR [13] at √ s = 52.6 GeV [4] and in pp interactions at the SppS collider [15] sizeable positive FB multiplicity correlations have been observed. Their strength was found to increase strongly with collision energy [3], which was confirmed later at much higher energies ( √ s 1 TeV) in pp collisions by the E735 collaboration at the Tevatron [12] and in pp collisions by the ATLAS experiment at the LHC ( √ s = 0.9 and 7 TeV) [16]. One of the observations reported by ATLAS is the decrease of b corr with the increase of the minimum transverse momentum of charged particles.
The STAR collaboration at RHIC analysed the FB multiplicity correlations in pp and Au-Au collisions at √ s NN = 200 GeV [17]. Strong correlation was observed in case of Au-Au collisions, while in pp collisions b corr was found to be rather small (∼0.1). In the present paper we relate this to the use of smaller pseudorapidity windows as compared to previous pp and pp measurements.
Forward-backward multiplicity correlations in high energy pp and A-A collisions also raise a considerable theoretical interest. First attempts to explain this phenomenon [7,[18][19][20] were made in the framework of the Dual Parton Model (DPM) [2] and the Quark Gluon String Model (QGSM) [21,22]. They provide a quantitative description of multiparticle production in soft processes. In improved versions of the models, collectivity effects arising due to the interactions between strings, which are particularly important in the case of A-A interactions, were taken into account [23][24][25][26]. These effects are based on the String Fusion Model (SFM) proposed in [27,28]. It was shown that these string interactions lead to a considerable modification of the FB correlation strength, along with the reduction of multiplicities, the increase of mean particle p T , and the enhancement of heavy flavour production in central A-A collisions [23,29,30].
FB correlations are usually divided into short and long-range components [2,7]. In phenomenological models, short-range correlations (SRC) are assumed to be localized over a small range of η-differences, up to one unit. They are induced by various short-range effects from single source fragmentation, including particles produced from decays of clusters or resonances, jet and mini-jet induced correlations.
The SFM predicts that the variance of the number of particle-emitting sources (strings) should be damped by their fusion, implying a reduction of multiplicity long-range correlations [23,25,26]. Contrary to this prediction, long-range correlations arising in the Color Glass Condensate model (CGC) [31] have been shown to increase with the centrality of the collision [32]. Therefore, the investigation of correlations between various observables, measured in two different, sufficiently separated η-intervals, is considered to be a powerful tool for the exploration of the initial conditions of hadronic interactions [33]. In the case of A-A collisions, these correlations induced across a wide range in η are expected to reflect the earliest stages of the collisions, almost free from final state effects [32,34]. The reference for the analysis of A-A collision dynamics can be obtained in pp collisions by studying the dependence of FB correlations on collision energy, particle pseudorapidity, azimuth and transverse momenta. This paper is organized as follows: Section 2 provides experimental details, including the description of the procedures used for the event and track selection, the efficiency corrections and systematic uncertainties estimates. Sections 3 and 4 discuss the results on FB multiplicity correlation measurements in η in pp collisions at √ s = 0.9, 2.76 and 7 TeV and in η-φ windows at √ s = 0.9 and 7 TeV. In Section 3, we present dependences of the correlation coefficient on the gap between windows, their widths and the collision energy. In Section 4, multiplicity correlations in windows separated in pseudorapidity and azimuth are studied, and the comparison with Monte Carlo generators PYTHIA6 and PHOJET is discussed. Results on multiplicity correlations in different p T ranges in pp collisions at √ s = 7 TeV are presented in Section 5.

Experimental setup, event and track selection
The data presented in this paper were recorded with the ALICE detector [35] in pp collisions at √ s = 0.9, 2.76 and 7 TeV. Charged primary particles are reconstructed with the central barrel detectors combining information from the Inner Tracking System (ITS) and the Time Projection Chamber (TPC). Both detectors are located inside the 0.5 T solenoidal field.
The ITS is composed of 3 different types of coordinate-sensitive Si-detectors. It consists of 2 silicon pixel innermost layers (SPD), 2 silicon drift (SDD) and 2 silicon strip (SSD) outer detector layers. The design allows for two-particle separation in events with multiplicity up to 100 charged particles per cm 2 . The SPD detector covers the pseudorapidity ranges |η| < 2 for inner and |η| < 1.4 for outer layers, acceptances of SDD and SSD are |η| < 0.9 and |η| < 1, respectively. All ITS elements have a radiation length of about 1.1% X 0 per layer. The ITS provides reliable charged particle tracking down to 0.1 GeV/c, ideal for the study of low-p T (soft) phenomena.
The ALICE TPC is the main tracking detector of the central rapidity region. The TPC, together with the ITS, provides charged particle momentum measurement, particle identification and vertex determination with good momentum and dE/dx resolution as well as two-track separation of identified hadrons and leptons in the p T region below 10 GeV/c. The TPC has an acceptance of |η| < 0.9 for tracks which reach the outer radius of the TPC and up to |η| < 1.5 for tracks that exit through the endcap of the TPC.
For the present analysis, minimum bias pp events are used. The minimum-bias trigger required a hit in one of the forward scintillator counters (VZERO) or in one of the two SPD layers. The VZERO timing signal was used to reject beam-gas and beam-halo collisions. The primary vertex was reconstructed using the combined track information from the TPC and ITS, and only events with primary vertices lying within ±10 cm from the centre of the apparatus are selected. In this way a uniform acceptance in  the central pseudorapidity region |η| < 0.8 is ensured. The data samples for √ s = 0.9, 2.76 and 7 TeV comprise 2 × 10 6 , 10 × 10 6 , and 6.5 × 10 6 events, respectively. Only runs with low probability to produce several separate events per one bunch crossing (so-called pile-up events) were used in this analysis.
To obtain high tracking efficiency and to reduce efficiency losses due to detector boundaries, tracks are selected with p T > 0.3 GeV/c in the pseudorapidity range |η| < 0.8. Employing a Kalman filter technique, tracks are reconstructed using space-time points measured by the TPC. Tracks with at least 70 space-points associated and track fitting χ 2 /n do f less than 2 are accepted. Additionally, at least two hits in the ITS must be associated with the track. Tracks are also rejected if their distance of closest approach (DCA) to the reconstructed event vertex is larger than 0.3 cm in either the transverse or the longitudinal plane. For the chosen selection criteria, the tracking efficiency for charged particles with p T > 0.3 GeV/c is about 80%.

Definition of counting windows
Two intervals separated symmetrically around η = 0 with variable width δ η ranging from 0.2 to 0.8 are defined as "forward" (F, η > 0) and "backward" (B, η < 0) . Correlations between multiplicities of charged particles (n) are studied as a function of the gap between the windows (denoted as η gap ). Another convenient variable is η sep which is the separation in pseudorapidity between centres of the windows. These variables are illustrated in Fig. 1, and all configurations of window pairs chosen for the analysis are drawn in Fig. 2.
The analysis is extended to correlations between separated regions in the η-ϕ plane (sectors). The ϕangle space is split into 8 sectors with the width δ ϕ=π/4 as shown in Fig. 3. This selection is motivated by a compromise between granularity and statistical uncertainty. The definitions and equations, described in Section 1, remain the same for the η-ϕ windows. The acceptance of the windows is determined by their widths δ η and δ ϕ as the ALICE acceptance is approximately uniform in the selected ranges of η and ϕ.

Experimental procedures of the FB correlation coefficient measurement
The present paper focuses on the study of FB correlation phenomena related to soft particle production. Therefore we restrict p T in 0.3 < p T < 1.5 GeV/c, except the study of the p T dependence presented in Section 5, where the p T range is 0.3 < p T < 6 GeV/c.
The correlation coefficients, b corr , for each window pair can be calculated using two methods. In the first method values of n B n F , n B , n F and n 2 F are accumulated event-by-event and then b corr is determined using Eq. 2. In the second method, b corr is calculated using linear regression. The 2-dimensional distributions (n B , n F ) are obtained integrating over all selected events, then the average backward multiplicity is calculated for each fixed value of the forward multiplicity, and b corr is obtained from a linear fit to the correlation function. It has been shown that the results obtained with the two methods agree within statistical uncertainty. In this work, results using the first method are presented.

Corrections and systematic uncertainties
Acceptance and tracking efficiency corrections are extracted from Monte Carlo simulations using PYTHIA6 [36] (Perugia 0 tune) and PHOJET [37,38] as particle generators followed by a full detector response simulation based on GEANT3 [39]. Corrections are done to primary charged particle correlations and multiplicities. Correction factors obtained with these two generators are found to agree within 1% and the difference is neglected. Three independent correction procedures are investigated.
In the first procedure, the correction factors for b corr are obtained as the ratio of b corr obtained at generator level (true value) to b corr after detector response simulation (measured value). In the second procedure the correction factors are obtained for n B n F , n B , n F and n 2 F separately and b corr is obtained from the corrected moments. The third procedure takes into account approximately linear dependence of b corr on n F when n F varies with cuts, and each corrected value of b corr is found by extrapolation to the corrected value of n F . It was found that results of all three procedures agree within 1.6-4.2% (see Table 1), thus proving the robustness of b corr determination. The second procedure was chosen as the most direct and commonly used to produce the final corrected value of b corr . Correction factors increase the values of b corr , obtained for standard cuts, by 6-10 % for analysis in η-windows and 9-18 % for analysis in η-φ windows and in p T intervals. By varying the selection cuts (vertex-, DCA-and track selection cuts), correction procedures, and by comparison of the high and low pile-up runs, the systematic uncertainties on b corr have been estimated. Adding all contributions in quadrature, the total systematic uncertainties are below 4.5% (4.2%, 3%) at √ s = 0.9 (2.76, 7) TeV for the b corr analysis in η-separated windows, and 6% for analysis in η-φ separated windows at √ s = 0.9 and 7 TeV. For the b corr analysis in p T intervals for 7 TeV, the systematic uncertainties are less than 8%. Statistical errors are small and within the symbol sizes for data  points in the figures. A summary of the contributions of systematic uncertainties for b corr in η-separated windows with the width δ η = 0.2 is presented in Table 1. Fig. 4 shows the FB multiplicity correlation coefficient b corr as a function of η gap and for different widths of the η windows (δ η) in pp collisions at the three collision energies. For each √ s, b corr is found to decrease slowly with increasing η gap , while maintaining a substantial pedestal value throughout the full η gap range.

Dependence on the width of windows
The δ η-dependence for adjacent (η gap = 0), symmetrical windows with respect to η=0 is shown in Fig. 5. For all collision energies, the correlation coefficient increases non-linearly with δ η. This trend is quite well described by PYTHIA6 and PHOJET, although the agreement worsens with increasing √ s. This δ η-dependence can be understood, along with other approaches [7,25,40], in a simple model with event-by-event multiplicity fluctuations and random distribution of produced particles in pseudorapidity. In this model, the multiplicity in an η interval containing the fraction p of the mean multiplicity N in the full η-acceptance is binomially distributed and its mean square is given by where N is the charged particle multiplicity measured in the pseudorapidity interval Y and One can connect the multiplicity fluctuations in the full η-acceptance considered in this analysis (Y = 1.6) with the correlation strength b corr (see Appendix A): Note that using Eq. 3 and Eq. 4 one can write the Eq. 5 also in the following form: From the measured ratio of the multiplicity variance σ 2 N ≡ N 2 − N 2 in Y = 1.6 to the mean value N we obtain the value of α at √ s = 0.9, 2.76 and 7 TeV to be 2.03, 3.25 and 4.42, respectively, with a systematic uncertainty of about 5%. The b mod corr (δ η)-dependences calculated by Eq. 5 are shown in Fig. 5 as red dashed lines. At η gap = 0 the b corr (δ η) dependence is well described by this simple model. However, this model is not able to describe the dependence of b corr on η gap in Fig. 4 because it does not take into account the SRC contribution mentioned above. Figure 4 shows that the pedestal value of b corr increases with √ s, while the slope of the b corr (η gap ) dependence stays approximately constant. This indicates that the contribution of the short-range correlations has a very weak √ s-dependence, while the long-range multiplicity correlations play a dominant role in pp collisions and their strength increases significantly with √ s. Note that this increase cannot be explained by the increase of the mean multiplicity alone. If, at different energies, we choose window sizes such that the mean multiplicity stays constant the increase is still observed (see Table 2).

Dependence on the collision energy
In the framework of the simple model described by Eq. 5 and 6 the increase of the correlation coefficient corresponds to the increase of the event-by-event multiplicity fluctuations with √ s characterized by the ratio σ 2 N / N . A strong energy dependence and rather large b corr values were previously reported by the UA5 collaboration [3] and recently by the ATLAS Collaboration [16]. However, as we see in Fig. 5   coefficient depends in a non-linear way on the width of the pseudorapidity window. One has to take this fact into account when comparing the correlation strengths obtained under different experimental conditions. In particular, it explains the small values of b corr observed by the STAR collaboration at RHIC (pp, √ s = 200 GeV) [17], where narrow FB windows (δ η = 0.2) were considered, while in previous pp and pp experiments wider windows of a few units of pseudorapidity were used.
Multiplicity correlations are also studied in different configurations of forward and backward azimuthal sectors. These sectors are chosen in separated forward and backward pseudorapidity windows of width δ η = 0.2 and δ ϕ = π/4 as shown in Fig. 3, resulting in 5 pairs with different ϕ-separation.
Figs. 6 and 7 show the azimuthal dependence of b corr as a function of different η sep , for 0.9 and 7 TeV, respectively. Data are compared to PYTHIA6 (tunes Perugia 0 and Perugia 2011), PHOJET and a parametric string model [41].
The string model fitted to our data helps to understand in a simple way the origins of the b corr behaviour. There are two contributions to b corr in this model. The short-range (SR) contribution originating from the correlation between particles produced from the decay of a single string and the long-range (LR) contribution arising from event-by-event fluctuations of the number of strings. The energy dependence of the fitted parameters demonstrates that SR parameters stay constant with √ s while the normalized variance of the number of strings, the only LR parameter of the model, increases by a factor of three. The 2-dimensional distribution of b corr as a function of η sep and ϕ sep is shown in Fig. 8 for √ s = 0.9 and 7 TeV. The qualitative behaviour of b corr resembles the results obtained for two-particle angular correlations: near-side peak and recoil away-side structure. The connection between the FB correlation and two-particle correlation function is discussed in detail in [7,[41][42][43].
The shape of the correlation function clearly indicate two contributions to the forward-backward multiplicity correlation coefficient. The SR contribution is concentrated within a rather limited region in the η-ϕ plane within one unit of pseudorapidity and π/2 in azimuth, while the LR contribution manifests itself as a common pedestal in the whole region of observation.
The strength of multiplicity correlations measured in η and η-ϕ windows is compared to the results obtained with PYTHIA6 [36] (tunes Perugia 0 and Perugia 2011) and PHOJET [37,38] Monte Carlo generators (MC). The detailed overview of key features of these generators can be found in [44]. Recent Perugia tunes for PYTHIA6 are described in [45]. In Fig. 9 the comparison of b corr as a function of η gap for δ η = 0.2 at √ s = 0.9, 2.76 and 7 TeV with the results obtained with different MC generators is shown. All models describe the data at √ s = 0.9 TeV reasonably well, while larger discrepancies are observed at 2.76 and 7 TeV, with PYTHIA giving a better description of the data than PHOJET. Qualitatively similar conclusions can be drawn from the comparison of the δ η-dependence in experimental data and MC as shown in Fig. 5.
Note that PYTHIA also describes the correlations in η-ϕ windows reasonably well, see Figures 6 and  7, while PHOJET gives a good description only for √ s = 0.9 TeV and significantly underestimates the data at 7 TeV.
The difference between the experimental data and the results obtained with MC generators is more visible in Fig. 10 as a function of η gap to MC calculations. The measured ratios show an increasing trend as a function of η gap , while PYTHIA and PHOJET underestimate the ratios and exhibit a flatter η gap dependence.
It is important to note that, in the framework of PYTHIA, the observed LR part of b corr (the pedestal in Fig. 8) is dominated by multiple parton-parton interactions (MPI). This supports earlier results [46], in which the FB correlations in pp collisions were studied by MC simulations with recent tunes of the PYTHIA6 at √ s = 0.9 TeV. Hence, the observed dependence of b corr on collision energy and on different configurations of rapidity and azimuthal windows adds new constraints on phenomenological models for multi-particle production.

Dependence of FB multiplicity correlation strength on the choice of p T intervals
The behaviour of FB multiplicity correlation strength was also studied as a function of p T of registered particles. These studies were motivated by a recent paper by the ATLAS collaboration [16], which reported a decrease in the multiplicity correlation strength with increasing p min T . However, as we have observed in Section 3.2, there is a strong non-linear dependence of b corr on the size of pseudorapidity windows and, hence, on the mean multiplicity n ch in the window (see Eq. 4, 5, and Fig. 5). In order to demonstrate that the strong p min T dependence is not a trivial multiplicity dependence, in our analysis we use p T intervals with the same n ch . To this end, the correlation strength b corr is studied for five p T intervals within 0.3 < p T < 6 GeV/c at √ s = 7 TeV: 0.3-0.4, 0.4-0.52, 0.52-0.7, 0.7-1.03 and 1.03-6.0 (GeV/c). In each p T interval, the corrected mean multiplicity n F = 0.157 with a systematic uncertainty about 2%. Correlations are studied in η and η-ϕ FB-windows configurations. Note that in case of windows chosen symmetrically with respect to η = 0 the definition of b corr given by (2) coincides with the correlation coefficient ρ n FB used in the ATLAS analysis. Fig. 11 shows b corr as a function of p min T for η gap = 0 and 1.2. Systematic uncertainties are shown as rectangles, statistical uncertainties are negligible. We find that b corr increases with p min T for both values of η gap , in contrast to the results reported in [16]. This result can be understood if one takes into account that the multiplicity fluctuations in a given window are closely connected with the two-particle correlation strength [7,43]. In the simple model with the event-by-event multiplicity fluctuations and random distribution of produced particles in pseudorapidity, discussed in Section 3.2, Eq. 7 allows us to discuss the observed dependence of the correlation coefficient b corr on the p T -binnings for the case of η gap = 0 (Fig. 11). One sees that the imposed condition n F =const eliminates the dependence of b corr on the multiplicity. The ratio 1/σ 2 n F decreases and b mod corr increases with increasing p min T . As mentioned above, in the approach used in [16] the dependence of the correlation strength on the p min T of charged particles was studied without cuts on p Tmax , which leads to a decrease of the correlation b corr with increasing p min T . This result can also be illustrated with the help of Eq. 7. In this case n F decreases with increasing p min T and n F /σ 2 n F increases (approaching the Poisson limit σ 2 n F = n F ) leading to the decrease of b mod corr . Thus, the difference of the results in these two approaches can be qualitatively understood using Eq. 7. Fig. 12 shows b corr as function of η gap for different p T intervals. Fig. 12 (a) compares data to PYTHIA6 tune Perugia 2011. The general trend of b corr increasing with higher p min T for all η gap is reproduced by this tune, with small quantitative deviations. Fig. 12 (b) shows the same data in comparison to PHOJET. This generator does not describe the data well: PHOJET results are almost independent of p min T and only grow significantly for the p T range 1.03-6.00 (GeV/c). Since experimental data was used to determine the p T intervals with the same mean multiplicity, the values of mean multiplicities may vary slightly in case of the MC samples for the same p T intervals. Deviations from the mean value are within 4% for PYTHIA6 Perugia 0 and 12% for PHOJET.
The analysis of b corr is also performed in η-ϕ separated windows in different p T intervals with the same mean multiplicity (for pp collisions at √ s = 7 TeV) in 8×8 η-ϕ windows. Results are shown in Fig. 13 and compared to PYTHIA6 and PHOJET calculations. In addition to the conclusions that were drawn above from the correlations between η-separated windows, some new details are revealed. In particular, one observes that the PHOJET discrepancy with the data is especially dramatic at ϕ sep =π/2, where PHOJET shows no dependence of b corr on the p T range. It was shown already in [47] that PHOJET has difficulties in description of underlying event measurements. Fig. 13 shows that for higher p T intervals a near-side peak appears (see panels for ϕ sep = 0 and π/4), at the same time the b corr in the flat region at η sep > 1 increases with p T for all ϕ sep values (compare panels for ϕ sep = π/2, 3π/4 and π). It should be emphasized that the value of the pedestal (the common constant component in all panels) increases with p T . In near-and away-side azimuthal regions the increase of b corr with p min T can be explained by an enhanced number of back-to-back decays and jets. The general rise of b corr can be related to the increase of the variance σ 2 N in Eq. 6, discussed in the framework of the simple model in Section 3.2.

Conclusion
The strengths of forward-backward (FB) multiplicity correlations have been measured in minimum bias pp collisions at √ s = 0.9, 2.76 and 7 TeV using multiplicities determined in two separated pseudorapidity windows separated by a variable gap, η gap , of up to 1.2 units. The dependences of the correlation coefficient b corr on the collision energy, the width and the position of pseudorapidity windows have been investigated. For the first time, the analysis has been also applied for various configurations of the azimuthal sectors selected within these pseudorapidity windows in events at √ s = 0.9 and 7 TeV.
A considerable increase of the FB correlation strength with the growth of the collision energy from √ s = 0.9 to 7 TeV is observed. It is shown that this cannot be explained by the increase of the mean multiplicity alone. The correlation strength grows with the width of pseudorapidity windows, while it decreases slightly with increasing pseudorapidity gap between the windows. It is shown that there is a strong non-linear dependence of the correlation strength on the width of the pseudorapidity windows and hence on the mean multiplicity value.
Measurements of the correlation strength for various configurations of azimuthal sectors enable the distinction of two contributions: short-range (SR) and long-range (LR) correlations. A weak dependence on the collision energy is observed for the SR component while the LR component has a strong dependence. For η-gaps larger than one unit of pseudorapidity and π/2 in azimuth the LR contribution dominates. This contribution forms a pedestal value (the common constant component) of b corr increasing with collision energy.
Moreover, pseudorapidity and pseudorapidity-azimuthal distributions of b corr have been obtained in pp events at √ s = 7 TeV for various particle transverse momentum intervals. It is found that the FB correlation strength increases with the transverse momentum if p T -intervals with the same mean multiplicity are chosen.
The measurements have been compared to calculations using the PYTHIA and PHOJET MC event generators. These generators are able to describe the general trends of b corr as a function of δ η, η gap and ϕ sep and its dependence on the collision energy. In p T -dependent analysis of b corr , PYTHIA describes data reasonably well, while PHOJET fails to describe b corr in azimuthal sectors. The observed dependences of b corr add new constraints on phenomenological models. In particular the transition between soft and hard processes in pp collisions can be investigated in detail using the p T dependence of azimuthal and pseudorapidity distributions of forward-backward multiplicity correlation strength b corr .

A A model with random uniform distribution of produced particles in pseudorapidity
In a simple model with event-by-event multiplicity fluctuations and random uniform distribution of produced particles in pseudorapidity, the multiplicity in an η interval containing the fraction p of the mean multiplicity N in the full η-acceptance is binomially distributed and its mean square is given by where N is the charged particle multiplicity measured in pseudorapidity interval Y and in the case of symmetric windows δ η F = δ η B = δ η One can rewrite (A.1)-(A.3) also as since the so-called robust variance R N is the same for any subinterval of Y in the case of the independent homogeneous distribution of the particles along Y [43].
Using the presentation for the covariance