Strongly intensive observable between multiplicities in two acceptance windows in a string model

Transcription

1 Strongly intensive observable between multiplicities in two acceptance windows in a string model Evgeny Andronov and Vladimir Vechernin Saint Petersburg State University 7/9 Universitetskaya nab. St.Petersburg 9934 Russia arxiv:88.977v [hep-ph] 9 Aug 8 Abstract The strongly intensive observable between multiplicities in two acceptance windows separated in rapidity and azimuth is calculated in the model with quark-gluon strings acting as sources. The dependence of this observable on the two-particle correlation function of a string the width of observation windows and the rapidity gap between them is analyzed. In the case with independent identical strings the model calculation confirms the strongly intensive character of this observable: it is independent of both the mean number of string and its fluctuation. or this case the peculiarities of its behaviour for particles with different electric charges are also analyzed. In the case when the string fusion processes are taken into account and a formation of strings of a few different types takes place in a collision this observable is proved to be equal to a weighted average of its values for different string types. Unfortunately in this case through the weight factors this observable becomes dependent on collision conditions and strictly speaking can not be considered any more as strongly intensive variable. or a comparison the results of the calculation of considered observable with the PYTHIA event generator are also presented Gz Particle correlations and fluctuations 3.85.Hd Inelastic scattering: many-particle final states Introduction It is known that the investigations of long range rapidity correlations give the information about the initial stage of high energy hadronic interactions []. So to find a signature of the string fusion and percolation phenomenon [ 3 4 5] in ultrarelativistic heavy ion collisions the study of the correlations between multiplicities in two separated rapidity intervals (known as the forward-backward multiplicity correlations was proposed [6]. Later it was realized [ ] that the investigations of the forward-backward correlations involving intensive observables in forward and backward observation windows as e.g. the event-mean transverse momentum enable to suppress the contribution of trivial socalled volume fluctuations originating from fluctuations in the number of initial sources (strings and to obtain more clear signal on the process of string fusion compared to usual forward-backward multiplicity correlations. In the present work we explore another way to suppress the contribution of volume fluctuations turning to the more sophisticated correlation observable. Basing on the multiplicities e.v.andronov@spbu.ru v.vechernin@spbu.ru

2 n and n B in two separated rapidity windows we study the properties of the so-called strongly intensive observable introduced in [4] where n B n ω n B + n B ω n cov(n n B n + n B ( cov(n n B n n B n n B ( and ω n and ω nb are the corresponding scaled variances of the multiplicities: ω n D n n = n n n. (3 In the framework of the model with color strings as particle emitting sources we calculate the dependence of the observable ( on the string two-particle correlation function the width of observation windows and the rapidity gap between them. We show that in the case with independent identical strings the strongly intensive character of this observable is being confirmed: it depends only on the individual characteristics of a string and is independent of both the mean number of strings and its fluctuation. We also analyze the peculiarities of the behaviour of the strongly intensive observables between multiplicities of particles with different electric charges and as well between multiplicities in two windows separated in rapidity and azimuth. In the case when the string fusion processes are taken into account and a formation of strings of a few different types takes place we found that this observable is equal to a weighted average of its values for different string types. Unfortunately in this case through the weight factors the observable becomes dependent on collision conditions and strictly speaking can not be considered any more as strongly intensive variable. We also present for a comparison the results of the calculation of considered observable with PYTHIA event generator. The paper is organized as follows. In Section we consider the most simple case with independent identical strings and symmetric π-azimuth observation windows separated by a rapidity gap. In Section 3 we generalize the obtained results for the case of two acceptance windows separated in rapidity and azimuth. Section 4 is devoted to the calculations of the strongly intensive observables between multiplicities of particles with different electric charges. In Section 4 the influence of the string fusion processes is analysed. Section 5 is devoted to the results obtained with the PYTHIA event generator. Σ in the model with independent identical strings We start our consideration from the simple case with symmetric π-azimuth observation windows δη = δη B δη separated by a rapidity gap η gap which corresponds to the distance η = η gap + δη between their centers. Clear that for symmetric reaction we have and the expression ( can be simplified to n = n B n ω n = ω nb ω n (4 n B = ω n cov(n n B / n = = D n cov(n n B n = n n n B n. (5 In the framework of the model with independent identical strings [5] we suppose that the number of strings N fluctuates event by event around some mean value N with some scaled variance ω N = D N / N. We expect that the intensive observables should not depend on N and the strongly intensive observables should not depend on both N and ω N. The fragmentation of each string contributes to the forward and backward observation rapidity windows δη and δη B the µ and µ B charged particles correspondingly which fluctuate

3 around some mean values µ and µ B with some scaled variances ω µ = D µ / µ and ω µb = D µb / µ B. Similarly to ( we can formally introduce also the Σ(µ µ B - the strongly intensive observable between multiplicities produced from decay of a single string: Σ(µ µ B µ ω µb + µ B ω µ cov(µ µ B µ + µ B. (6 or symmetric reaction and symmetric observation windows it also can be simplified to µ = µ B µ ω µ = ω µb ω µ (7 Σ(µ µ B = ω µ cov(µ µ B / µ = = D µ cov(µ µ B µ = µ µ µ B µ. (8 Clear that in this model n = N µ = N µ δη ω n = ω µ + ω N µ (9 so we see that the ω n is intensive but not strongly intensive observable. We also supposed the translation invariance in rapidity where µ is a distribution density for particles produced from a single string. or us it is important to remember that the scaled variance ω n of the number of particles produced in the rapidity interval δη is determined by the two-particle correlation function C (η η [6 7]: ω n = + n I ( where I δη dη dη C (η η. ( δη δη Here the two-particle correlation function is defined by the standard way (see e.g. [8]: and ρ(η dn ch dη C (η η ρ (η η ρ(η ρ(η ( ρ (η η d N ch dη dη. (3 In the case with a translation invariance in rapidity we have the uniform distribution ρ(η = ρ ρ (η η = ρ (η η (4 and the correlation function C (η η depending only on a difference of rapidities. Similarly we can also introduce the two-particle correlation function of a single string Λ(η η λ (η η λ(η λ(η (5 for description of the correlation between particles produced from a same string where λ(η and λ (η η are the corresponding single and double distributions. or the translation invariant case λ(η = µ λ (η η = λ (η η (6 and similarly we have where ω µ = + µ J (7 J δη dη dη Λ(η η (8 δη δη 3

4 and the string two-particle correlation function Λ(η η depends only on a difference of rapidities. Note that by formula (7 we see that the so-called robust variance [8]: R µ ω µ µ = J (9 and depends only on a string correlation function Λ(η η. The similar formulae are valid for the corresponding covariances [7]: cov(n n B n n B = I B ( where I B δη dη dη C (η η ( δη δη B where the integration over η and η are fulfilled now on different rapidity intervals δη and δη B correspondingly. or one string we also have cov(µ µ B µ µ B = J B ( where J B δη dη dη Λ(η η (3 δη δη B In the considered model with independent identical strings we can express the observable correlation function C (η η through the correlation function of a single string Λ(η η [7]: C (η η = Λ(η η + ω N N (4 What leads immediately to I B = J B + ω N N I = J + ω N N (5 and for symmetric forward-backward windows ω n = D n / n = + µ [J + ω N ] (6 cov(n n B / n = µ [J B + ω N ] (7 We took also into account that ρ = N µ. Then by (5 ( and ( we obtain n B = + n [I I B ] = = + µ [J J B ] = Σ(µ µ B (8 where the Σ(µ µ B is the strongly intensive observable between multiplicities produced from decay of a single string defined by (6 and (8. By (8 we really see that in the framework of this model the observable n B is a strongly intensive it is independent of both the mean number of string N and its fluctuation ω N. It depends only on the string parameters µ Λ(η η and the width of observation windows µ = µ δη. Whereas the scaled variance ω n is an intensive but not a strongly intensive observable because it is independent on the mean number of string N but through ω N depends on their fluctuation. We should note that other quantities that characterize correlations between multiplicities in two windows such as a correlation coefficient [9 ] or a variance of asymmetry [] are also not strongly intensive and therefore are more sensitive to experimental event selection procedures. 4

5 Table : The value of the parameters in formula (3 [7] for the two-particle correlation function of a string Λ( η φ fitted by the experimental pp ALICE data on forward-backward correlations between multiplicities in windows separated in rapidity and azimuth at three initial energies [] together with the value of scaled variance of the number of strings ω N. s TeV LRC µ ω N.7.4. µ Λ η φ..5. SRC µ Λ η... φ η or small observation windows of a width δη η corr where the η corr is the characteristic correlation length for particles produced from the same string the formulae (6-8 takes especially simple form: ω n = D n / n = + µ δη [Λ( + ω N ] (9 cov(n n B / n = µ δη [Λ( η + ω N ] (3 n B = + µ δη [Λ( Λ( η] = Σ(µ µ B (3 where η = η η B is a distance between the centers of the forward an backward observation windows. rom the last formula we see the main properties of the n B which we expect in this model. Starting from the value it increases with a distance η between the centers of the observation windows since the two-particle correlation function of a string Λ( η decrease with η. The extent of the Σ( η increase with η is proportional to the width of the observation windows δη. More detailed description of the n B needs the knowledge of the two-particle correlation function of a string Λ( η. In paper [7] in the framework of the model with independent identical strings this function was fitted using the experimental pp ALICE data on forwardbackward correlations between multiplicities in windows separated in rapidity and azimuth at three initial energies together with the value of scaled variance of the number of strings ω N (see table : where it was implied that Λ( η φ = Λ e η η e φ ϕ + ( +Λ e η η η + e η+η η e ( φ π ϕ (3 φ π. (33 or φ > π one must periodically extend Λ( η φ to φ φ+πk. With such completion the Λ( η φ meets the following requirements: Λ( η φ = Λ( η φ (34 Λ( η φ = Λ( η φ (35 Λ( η φ + πk = Λ( η φ. (36 Recall that the comparison of the model with experimental data in [7] enables to fix only the product of the parameters µ Λ µ Λ and µ ω N. 5

6 TeV.76 TeV.9 TeV exponent.fits.7 µ ( igure : The two-particle correlation function of a string Λ( η (integrated over azimuth obtained by a fitting [7] of the experimental pp ALICE data [] on forward-backward correlations between multiplicities at three initial energies:.9.76 and 7 TeV (the dashed lines and the corresponding exponential fits (39 (solid lines with the parameters presented in table. Table : The value of the parameters in formula (39 for the two-particle correlation function of a string Λ( η obtained by a fitting [7] of the experimental pp ALICE data [] on forward-backward correlations between multiplicities at three initial energies (see ig.. s TeV µ Λ η corr Our two-particle correlation functions ( and (5 defined for π-azimuth observation windows can be obtained by simple integration over azimuth: C ( η = π Λ( η = π π π C ( η φ d φ (37 Λ( η ϕ d ϕ. (38 So by integration of the fit (3 we find the Λ( η presented in ig.. The obtained dependencies in this figure for three initial energies are well approximated by the exponent Λ( η = Λ exp( η /η corr (39 with the parameters presented in table. We see that the correlation length η corr decreases with the increase of collision energy. This can be interpreted as a signal of an increase with energy of the admixture of strings of a new type - the fused strings in pp collisions (see below. The results of the calculation of the strongly intensive observable n B by formulae (8 with this two-particle correlation function for two width of the observation windows δη =. and.4 are presented in ig. for three initial energies:.9.76 and 7 TeV. By formulae (5 and (9-3 we can understand the behaviour of n B in this figure as follows. The formula (5 shows that for symmetric reaction and symmetric observation windows the n B is proportional to the difference between the variance D n and the covariance 6

7 δη=.4 B n..5. δη=..5 pp δη=.4 7. TeV.76 TeV.9 TeV δη=. 7. TeV.76 TeV.9 TeV distance between window centers - η igure : The strongly intensive observable n B between multiplicities in two small pseudorapidity windows (of the width δη =. and.4 as a function of the distance between window centers η calculated in the model with independent identical strings using the two-particle correlation function of a string Λ( η (see ig. obtained by a fitting [7] of the experimental pp ALICE data [] on forward-backward correlations between multiplicities at three initial energies:.9.76 and 7 TeV. cov(n n B. It is important to remember that the value both of them are determined by the string two-particle correlation function Λ and the scaled variance in the number of strings ω N (see formulae (9 and (3. In particular in the absence of correlations between particles produced from a given source the multiplicity distribution from such source will be poissonian (ω µ = see formula (7. irst of all we see that in n B which by (5 is a difference between D n / n and cov(n n B / n the contributions from the variance in the number of strings ω N are being mutually canceled (see formulae (6 (7 or (9 (3 what reflects the strongly intensive character of the quantity. Moreover by (9 we see that at small values of the distance between observation windows η η corr the contribution µ δηλ( of the two-particle correlations to ω n is being compensated by their contribution µ δηλ( η to cov(n n B / n and the n B is equal to. At large distances between observation windows η η corr by formula (39 the twoparticle correlation function of a string Λ( η goes to zero and the n B saturates to ω µ = + µ δηλ(. So we have n B at η η corr (4 n B ω µ at η η corr. (4 7

8 .9 TeV δη=. δϕ=.5π.76 TeV δη=. δϕ=.5π n B φ.5 η n n B φ.5 η n 7 TeV δη=. δϕ=.5π n B φ.5 η n igure 3: The strongly intensive observable n B between multiplicities in two windows of the width δη =. and δφ = π/4 as a function of the distance between window centers η in rapidity and φ in azimuth calculated in the model with independent identical strings using the two-particle correlation function of a string Λ( η φ (3 obtained by a fitting [7] of the experimental pp ALICE data [] on forward-backward correlations between multiplicities at three initial energies:.9.76 and 7 TeV in ALICE TPC pseudorapidity acceptance. Note that ω µ increases with the width δη of the observation windows (9. In ig. we see also some general increase of the n B with initial energy below in Section 5 we will show that in the framework of the string model it can be interpreted as a signal of an increase with energy of the admixture of strings of a new type - the fused strings in pp collisions. 3 Σ for windows separated in rapidity and azimuth All results obtained in Section can be easily extended to the case of the strongly intensive observable n B between multiplicities in two acceptance windows separated both in rapidity and azimuth. In particular for symmetric reaction and symmetric small observation windows of the width δη in rapidity and δφ in azimuth with the separation between their centers η sep = η and φ sep = φ we find in the model with independent identical strings: n B = + δη δφ π µ [Λ( Λ( η φ] (4 which is a generalization of the formula (3. If we use now again for the two-particle correlation function of a string Λ( η φ the approximation (3 suggested in paper [7] with the parameters (see table fitted by the experimental pp ALICE data [] on forward-backward correlations between multiplicities in windows separated in rapidity and azimuth at three initial energies then we find the behaviour of n B presented in ig.3 in ALICE TPC acceptance and in ig.4 extrapolated to a wider rapidity interval. The explanation of this behaviour of n B on the base of the formula 4 is absolutely the same as in the end of the Section. 8

9 .9 TeV δη=. δϕ=.5π.76 TeV δη=. δϕ=.5π n B.... n B n.... n φ 3 4 η φ 3 4 η 7 TeV δη=. δϕ=.5π n B.... n φ 3 4 η igure 4: The same as in ig.3 but with π/8 azimuth window width and extrapolated to a wider interval of the separation between windows in rapidity: the η is upto 4 rapidity units. 4 Σ with charges In Section we have introduced the strongly intensive observable Σ based on multiplicities of the all charged hadrons measured in two pseudorapidity intervals. Now we consider different combinations of electric charges in these windows and similarly to formula ( we define + n+ B n B + n B and n+ B. or a symmetric reaction and symmetric observation windows we have n + = n+ B n+ ω n + and the same for n. In this case we have also and the definitions can be reduced to = ω n + ω n + (43 B cov(n + n = cov(n+ B n B (44 cov(n + n B = cov(n n+ B (45 + n+ B = ω n + cov(n+ n+ B n + n B = ω n cov(n n B n (46 (47 + n B = n+ B = = n+ ω n + n ω n + cov(n + n B n. (48 We can also introduce an additional strongly intensive observable that measures correlation between multiplicities of different charges in the same window [6]: + n = + B n B = = n+ ω n + n ω n + cov(n + n n. (49 9

10 By expanding n = n + + n and n B = n + B + n B in (5 and taking into account (46-(49 we find the following elegant relation: and n B = n+ n + n+ B + n n n B n B + n. (5 This relation can be further simplified in case of charge symmetry when n + = n = n / (5 ω n + = ω n cov(n + n+ B = cov(n n B what is a very good approximation for mid-rapidity region at LHC collision energies. In this case we have n B = + n+ B + + n B + n. (5 In order to calculate charge-dependent strongly intensive observables in the model of independent strings we have to define corresponding one- and two-particle distributions describing decay properties of a source. or the charge symmetry case we have: λ + (η = λ (η = λ(η (53 Λ ++ (η η = Λ (η η (54 Λ + (η η = Λ + (η η (55 Λ(η η = ( Λ ++ (η η + Λ + (η η. (56 Using the translation invariance in rapidity it is again conveniently to define the following quantities: J ++ δη dη dη Λ ++ (η η (57 δη δη J + δη dη dη Λ + (η η (58 δη δη J ++ B δη dη dη Λ ++ (η η (59 δη δη B J + B δη dη dη Λ + (η η. (6 δη δη B Then by (8 (3 (53 (56 and (57-6 we have What leads to the following relations: or in the case of small windows: µ + = µ = µ = µ δη (6 J = ( J ++ + J + (6 J B = ( J ++ B + J + B. (63 + n+ B = + µ+ (J ++ J ++ B (64 + n B = + µ+ (J ++ J + B. (65 + n = + µ+ (J ++ J + (66 + n+ B = + µ δη[λ ++ ( Λ ++ ( η] (67

11 + n B = + µ δη[λ ++ ( Λ + ( η] (68 + n = + µ δη[λ ++ ( Λ + (]. (69 We see that as expected + n+ B at η however the + n B tends to be equal to + n in this limit which is not necessarily equal to. To deduce how + n B behaves at small η one need additional input from experiments. Looking at relations (67-69 one can immediately notice certain similarities with chargedependent correlations measured via the so-called balance function B ( η [3]. In this paper the balance function is defined to be proportional to the difference between unlike-sign and like-sign two-particle correlations functions: or simply B( η φ [C + + C + C ++ C ] (7 B( η φ = C + C ++. (7 The last equation exploits the charge symmetry in mid-rapidities at LHC energies. rom (4 we expect that the B( η is proportional to Λ + ( η Λ ++ ( η i.e. to + n B + n+ B. Really taking into account the normalization of the two-particle correlations functions used in paper [3] we find B proj ( η = 4 µ [Λ + ( η Λ ++ ( η]. (7 Here following [3] the pseudorapidity dependence of balance function is defined as a projection of two-dimensional B ( η φ: B proj ( η 3π π B( η φ d φ. (73 Note that as it is pointed out before the subsection 6.. in the paper [3] the results for B proj ( η will be two times larger if calculating the two-particle correlation functions entering the definition of balance function (7 one will not impose the requirement that the transverse momentum of the trigger particle must be higher than the associated one i.e. we would have coefficient / in (7 instead of /4 for the balance function normalized in such a way. So by fitting the experimental pp ALICE data on balance functions [3] and keeping in mind relation (56 one can extract parameters of unlike-sign and like-sign correlation functions Λ + ( η and Λ ++ ( η and in turn predict η dependencies of the variables + n+ B and + n B. or better data fitting we have to take into account the HBT correlations for like-sign twoparticle correlation function and complement the form of parametrization (39: We keep it unmodified for unlike-sign correlations Λ ++ ( η = Λ ++ exp ( η /η ++ + [ ( +Λ HBT exp η/η HBT ]. (74 Λ + ( η = Λ + exp ( η /η +. (75 Here we assumed also that HBT-correlations appears only for pairs of particles originating from the same string. We perform simultaneous fitting of the experimental data on pp collisions at 7 TeV for balance functions [3] and of µ Λ ( η extracted from forward-backward correlations [] (see In order to take into account decays of neutral resonances one need to add a characteristic contribution to the unlike-sign correlations function. We postpone this modification for future research.

12 p+p 7TeV 7-8% centrality p+p 7TeV minimum bias ( η proj B...8 fit data Λ( η c cor µ.4. fit model parametrization η η igure 5: Left: The projection of balance function B proj ( η as a function of the distance between two particles η measured by the ALICE experiment [3] for 7-8% centrality class in pp collisions at 7 TeV together with the fit (7 obtained using the difference between the unlike-sign and like-sign two-particle correlation functions of a string Λ + ( η Λ ++ ( η. Right: The two-particle correlation function of a string ig. corrected for p T acceptance (see text for all charged particles extracted [7] from the experimental pp ALICE data on forward-backward correlations between multiplicities at 7 TeV [] together with the fit (56 obtained using the sum of the unlike-sign and like-sign two-particle correlation functions of a string Λ + ( η+λ ++ ( η. ig.. As the forward-backward correlations were measured experimentally for minimum bias pp events the results on balance functions for 7-8% pp centrality class were selected assuming that the minimum bias is dominated by the peripheral collisions. Moreover in order to compensate the more narrow transverse momentum interval taking into account in -B correlations measurements compared to balance function investigations ((.3;.5 GeV/c in [] and (.; GeV/c in [3] we multiply extracted µ Λ ( η by a correction factor c cor =.8 that was estimated in the PYTHIA model [4 5] as a ratio of mean multiplicities at mid-rapidity for corresponding p T intervals. igure 5 shows comparison of experimental data for 7-8% centrality class in pp collisions at 7 TeV with the suggested fit with parameters being listed in Table 3. igure 6 shows + n B + n+ B and n B dependencies on η obtained with parameters listed in Table 3. All functions show growing behaviour with η with decreasing difference between unlike-sign and like-sign strongly intensive observables. This is a consequence of B ( η at large η. Unlike-sign Σ is smaller than at small η becoming greater than at larger η. Like-sign Σ shows behaviour that is similar to any charge sign case (see ig. but suppressed in absolute value. Note that n B was calculated here by (5. It rises slightly faster than in igure because p T interval was rescaled. In Table 3 we see also that as one can expect from the local charge conservation in string fragmentation process [7] the correlation length η ++ for the particles of same charges is larger than the one η + for opposite charges.

13 Table 3: The value of the parameters in formulae (7475 for the two-particle correlation functions of a string Λ + ( η and Λ ++ ( η obtained by a simultaneous fitting of the experimental ALICE data on balance function (B [3] and on forward-backward correlations (BC between multiplicities [] for pp collisions at 7 TeV (see ig. 5. s TeV 7. µ Λ +.4 µ Λ η +.34 η µ Λ HBT.5 η HBT.33 p+p 7TeV Σ n n + B - B n B igure 6: The strongly intensive observables + n B (dashed + n + B (dashdotted and n B (solid as a function of the distance η between two intervals of width δη =. calculated in the model with independent identical strings using the unlike- and like-sign two-particle correlation functions of a string Λ + ( η and Λ ++ ( η. 5 Σ with string fusion In this section we consider the influence of processes of interaction between strings on the strongly intensive observable n B. This influence increases with initial energy and with going from pp to heavy ion collisions. One of the possible ways to take these processes into account is to pass from the model with independent identical strings to the model with string fusion and percolation [ 3 4 5]. Technically to simplify the account of the string fusion processes one can introduce the finite lattice (the grid in the impact parameter plane. This approach was suggested in [8] and then was successfully exploited for a description of various phenomena (correlations anisotropic azimuthal flows the ridge in ultra relativistic nuclear collisions [ ]. In this approach one splits the impact parameter plane into cells which area 3 η

14 is equal to the transverse area of single string and supposes the fusion of all strings with the centers in a given cell. This leads to the splitting of the transverse area into domains with different fluctuating values of color field within them. What is similar to the attempts to take into account the density variation in transverse plane in models based on the BKL evolution [35] and on the CGC approach [36]. In this model the definite set of strings of different types corresponds to the given event. Each such string originating from a fusion of k primary strings is characterized by its own parameters: the mean multiplicity per unit of rapidity µ (k and the string correlation function Λ k ( η. By (8 these parameters uniquely determine the strongly intensive observable between multiplicities produced from decay of a string of the given type Σ k (µ µ B defined by (6 and (8. or example for two small observation windows δη η corr (k separated by the rapidity distance η similarly to (3 we have Σ k (µ µ B = + µ (k δη [Λ k( Λ k ( η]. (76 In this case of the model with k string types the direct calculation gives for the strongly intensive observable n B (for symmetric reaction and observation windows δη = δη B δη n = n B n : n B = k= α k Σ k (µ µ B α k = n(k n (77 where n (k is a mean number of particles produced from all sells with k fused strings in the observation window δη. Note that the same result was obtained in the model with two types of strings in [37] for the long-range part of n B when at η η corr by (8 and (3 we have J J B and by (7 and (8 Σ k (µ µ B = ω (k µ with k =. What led to n B η ηcorr = n ω( µ + n ω µ ( n. (78 One can compare this limit with the one given by the formula (4 for the case of identical strings. or an arbitrary rapidity distance η between the forward and backward observation windows of the δη width in the model with k string types we have n B = + δη k= α k µ (k [J (k J (k B ] (79 where we have introduced J (k (k and J B for the two-partic-le correlation function Λ k(η η similarly to (8 and (3. or narrow observation windows δη η corr (k by (76 it is simplified to n B = + δη k= α k µ (k [Λ k ( Λ k ( η]. (8 If we will use also the simple exponential parametrization for Λ k ( η similar to (39: then we can rewrite (8 as Λ k ( η = Λ (k exp ( η /η (k corr (8 n B = + δη α k µ (k Λ(k [ exp ( η /η(k corr]. (8 k= 4

15 We see that in this case each string of the type k is characterized by two parameters: the product µ (k Λ(k where the µ(k is the mean multiplicity per unit of rapidity from a decay of such string and its two-particle correlation length η corr (k which determines the correlations between particles produced from a decay of the string. In the framework of the string fusion model [ 3 4 5] one usually supposes that the mean multiplicity per unit of rapidity for fused string µ (k increase as k with k. The dependence of the correlation length η corr (k on k is not so obvious. Basing on a simple geometrical picture of string fragmentation (see e.g. [ ] one can expect the decrease of the correlation length η corr (k with increase of k. In this picture with a growth of string tension the fragmentation process is finished at smaller string segments in rapidity. The correlation takes place only between particles originating from a fragmentation of neighbour string segments and hence the correlation length η corr (k will decrease with k for fused strings. Indirectly this fact is confirmed by the analysis [4] of the experimental STAR [43] and ALICE [44] data on net-charge fluctuations in pp and AA collisions. The dependence of netcharge fluctuations on the rapidity width of the observation window can be well described in a string model if one supposes the decrease of the correlation length with the transition to collisions of heavier nuclei and to higher energies i.e. to collisions in which the proportion of fused strings is increasing. By (8 both these factors the increase of µ (k and the decrease of η (k corr for fused string lead to the steeper increase of Σ k (µ µ B (76 with η and to its saturation at a higher level ω µ (k = + δη µ (k Λ k(. Due to (77 this behaviour transmits to the observable n B as the last is a weighted average of Σ k (µ µ B with the weights α k = n (k / n which are the mean portions of the particles produced from a given type of strings. In real experiment we have always a mixture of fused and single strings. So with the transition to pp collisions at higher energy or/and to collisions of nuclei the proportion of fused strings will increase and we will observe the steeper increase of n B with η and its saturation at a higher level. Really in ig. we see such behaviour of n B when we compare n B for pp collisions at three initial energies:.9.76 and 7 TeV obtained through fitting [7] of the experimental pp ALICE data [] on forward-backward correlations between multiplicities. Table illustrates the increase of µ Λ and the decrease of the correlation length η corr with energy for this data. Note that these values are the some effective ones because at each energy we had supposed that all strings are identical. So they only indirectly reflects the influence of the increase of the proportion of fused strings with energy in pp collisions. or studies of the n B dependence on multiplicity classes we can predict the behaviour similar to the one in ig.. or more central pp collisions due to the increase of the proportion of fused strings in such collisions we also have to observe the steeper increase of n B with η and its saturation at a higher level. Note that from a general point of view this simultaneously means that the observable n B strictly speaking can not be considered any more as strongly intensive. Through the weight factors α k = n (k / n entering the formula (77 which are the mean portions of the particles produced from a given type of strings the observable n B becomes dependent on collision conditions (e.g. on the collision centrality. 6 Σ with PYTHIA In this section for a comparison we present the results for the strongly intensive observables under consideration obtained using the PYTHIA event generator. The PYTHIA event generator [4 5] with the Lund string fragmentation model [45] in its core is very successful in description of LHC data on pp collisions. So one can expect that the results will be in correspondence with the ones obtain above in a simple string model. All the results to be shown below are obtained with the PYTHIA8.3 version using its 5

16 n B. Pythia8..9 TeV.76 TeV.9 7 TeV η igure 7: The strongly intensive observable n B between multiplicities of charged particles with.3 < p T <.5 GeV/c in two small pseudorapidity windows (of the width δη =. as a function of the distance between window centers η calculated with the Monash 3 tune of the PYTHIA8.3 model for three collision energies:.9.76 and 7 TeV. default Monash 3 tune [46] from generation of 6 events for all inelastic protonproton collisions (SoftQCD:inelastic=on. Statistical uncertainties were estimated using the sub-sample method [47] with number of sub-samples being equal to 3. Statistical uncertainties are smaller than a line width and not visible on the plots to be presented below. In the analysis we considered only charged particles with.3 < p T <.5 GeV/c to be consistent with the ALICE forward-backward correlations measurements []. igure 7 shows PYTHIA8 predictions of n B dependencies on distance between two windows for three collision energies. We see that n B grows with separation of windows up to a certain saturation point with a subsequent decrease. The point of saturation increases with the collision energy growth. Moreover for η < the growth is steeper than for large gaps. The phase of growing is reminiscent of the independent string model predictions (see ig.. Observed collision energy dependence is also consistent with the predictions from Sect.. Change of trend at large separation between windows can be understood as a consequence of a significant decrease of a mean multiplicities n and n B due to reduction of the number of strings contributing to both observation windows. Such a decrease leads to almost poissonian fluctuations of n and n B and consequently n B. This effect is not taken into account in the independent string model where at mid-rapidities we assume that the n and n B are independent of window positions due to translation invariance in rapidity. We supposed also that every string can contribute to both observation windows. igure 8 shows results for different combinations of electric charges. Additional point with η = is calculated for + n B in the same window ( = B see formulae (68 and (69. The behavior of + n B + n+ B and n B is in correspondence with the independent string model predictions (see igure 6. Unlike-sign Σ starting also from the value around.95 at small η becomes greater than at large η. Like-sign Σ shows behaviour 6

17 n B. Pythia8..9 n n n Σ ( B + B B + + η+ Σ ( η Σ ( η igure 8: The strongly intensive observable n B between multiplicities of charged particles with.3 < p T <.5 GeV/c in two small pseudorapidity windows (of the width δη =. as a function of the distance between window centers η for.9 TeV collisions calculated with the Monash 3 tune of the PYTHIA8.3 model for different charge combinations. or comparison the results obtained by the formula (5 are shown in full circles. that is similar to all charged case n B but suppressed in absolute value. ull circles in igure 8 represent results obtained in PYTHIA using the relation (5. Recall that this relation was obtained under the assumption of charge symmetry (5. Really one can see that this relation nicely reproduces n B for all charged particles (solid line in central rapidity region where the charge symmetry take place at LHC energies while with going to a fragmentation region it starts to fail and two curves begin to deviate. Results obtained with the PYTHIA8 event generator for n B for windows separated both in pseudorapidity and azimuthal angle are presented in ig. 9. The shape of obtained functions - a dip at small values of η and φ and a plateau at larger values - is again in qualitative agreement with the independent string model predictions (see ig. 3. It is important to note that in the framework of the simple string model in contrast with PYTHIA calculations we can clearly see the physical reasons for such behavior. 7 Summary and conclusions The using of strongly intensive observables are considered as a way to suppress the contribution of trivial volume fluctuations in experimental studies of the correlation and fluctuation phenomena (see e.g. [4]. In present paper we have studied the properties of strongly intensive observable between multiplicities in two acceptance windows separated in rapidity and azimuth n B in the model with quark-gluon strings (color flux tubes as sources. We show that in the case with independent identical strings the strongly intensive character of this observable is being confirmed: it depends only on the individual characteristics of a string and is independent of both the mean number of strings and its fluctuation. These individual 7

18 .9 TeV.3<p <.5 GeV/c δη=. δϕ=π/4 T.76 TeV.3<p <.5 GeV/c δη=. δϕ=π/4 T n B.3 n B ϕ.5 η 3 3 ϕ.5 η 7 TeV.3<p <.5 GeV/c δη=. δϕ=π/4 T n B ϕ.5 η igure 9: The strongly intensive observable n B between multiplicities in two small pseudorapidity - azimuthal angle windows (of the width δη =. δϕ = π/4 as a function of the distance between window centers η ϕ calculated with the Monash 3 tune of the PYTHIA8.3 model for three collision energies:.9.76 and 7 TeV. characteristics of a string are a mean number of particles per unit of rapidity µ produced from string fragmentation and the two-particle correlation function Λ( η φ characterizing the correlations between particles produced from the same string. The ALICE experimental data [] on forward-back-ward correlations (BC in small rapidity windows separated in rapidity and azimuth enables to obtain the information on this string correlation function Λ( η φ [7]. Using it we calculate the dependence of the strongly intensive observable n B on the acceptance of observation windows and the gaps between them. We have studied also the strongly intensive observables between multiplicities taking into account the sigh of particle charge + n+ B + n B and + n. We express them through the string correlation functions between like and unlike charged particles Λ ++ ( η and Λ + ( η. To calculate these quantities we need more information on a string decay process because the BC data [] contains information only on the sum of these correlation functions Λ( η = [Λ ++ ( η + Λ + ( η]/. We show that the so-called balance function (B can be expressed through the difference of these string correlation functions. Using the ALICE experimental data [3] on B we extract both correlation functions between like and unlike charged particles produced from a fragmentation of a single string. With these correlation functions we calculate the properties of the strongly intensive observables + n+ B + n B and + n. In particular we found that as one can expect from the local charge conservation in string fragmentation process [7] the correlation length for the particles of same charges is larger than the one for opposite charges. In the case when the string fusion processes are taken into account and a formation of strings of a few different types takes place in a collision we show that the observable n B is proved to be equal to a weighted average of its values for different string types. Unfortunately in this case through the weight factors this observable becomes dependent on collision conditions and strictly speaking can not be considered any more as strongly intensive variable. This complicates the quantitative predictions for the n B as it starts to depend on experimental conditions through these weight factors. Nevertheless we argue that the string fusion leads to the following changes of individual 8

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