Short- and long-range rapidity correlations in the model with a lattice in transverse plane

In the framework of the quark-gluon string model we consider the various fluctuation and correlation observables used in the analysis of the multiparticle production in hadronic interactions at high energy. We express these observables through the fundamental string characteristics and analyze their resulting properties: the dependence on the width of observation window(s), the range of the correlation in rapidity, the intensive or strongly intensive behavior. To take into account the influence of a string fusion processes on the string characteristics and on the behavior of the observables. we use the version of string model with a lattice (grid) in the impact parameter plane. In particular we show that the observable between multiplicities in two acceptance windows separated in rapidity, which is a strongly intensive in the case with independent identical strings, loses this property, when we take into account the string fusion and the formation of strings of a few different types takes place in a collision.

Short-and long-range rapidity correlations The correlation coefficient: A. Capella and A. Krzywicki, Phys.Rev.D18, 4120 (1978) The locality of strong interaction in rapidity ⇒ Short-Range FB Correlations (SRC) (between particles from a same string) Event-by-event variance in the number of cut pomerons ( Strongly influenced by "volume" fluctuations.
The b nn is connected with two-particle correlation function C 2 , canonically defined as are the single and double inclusive particle distributions. For small δη F -δη B observation windows we have: We have used that for small windows: D n F ≈ n F . Also influenced by "volume" fluctuations.
To suppress the influence of trivial "volume" fluctuations we have to go from traditional extensive variables n F and n B to new intensive variables, e.g. event-mean transverse momenta p F and p B of all particles (n F and n B ) in the intervals δη F and δη B (see e.g. [V.V., EPJ Web of Conf. 125, 04022 (2016)]) OR to study more sophisticated correlation observables, e.g. the strongly intensive observable Σ(n F , n B ). where and ω n F and ω n B are the corresponding scaled variances of the multiplicities: and Connection with FBC coefficient b nn : Valday (May 29, 2018) QUARKS-2018 V. Vechernin 7 / 40 Σ(n F , n B ) through two-particle correlation function C 2 ω n = D n / n = 1 + n I FF , cov(n F , n B )/ n = n I FB , (7) where The last limit is valid for the small windows: δη F =δη B =δη η corr , then The model with independent identical strings [M.A. Braun, C. Pajares, V.V.V., Phys. Lett. B 493, 54 (2000)] 1) The number of strings, N, fluctuates event by event around some mean value, N , with some scaled variance, Intensive observable does not depends on N . Strongly intensive observable does not depends on N and ω N .
2) The fragmentation of each string contributes event-by-event to the forward and backward observation rapidity windows, δη F , and δη B , the µ F and µ B charged particles correspondingly, which fluctuate around some mean values, µ F and µ B , with some scaled variances, The observation rapidity windows are separated by some rapidity interval: η sep = ∆η -the distance between the centers of the δη F and δη B .
Clear that in this model (and the same for n B ): Two-particle correlation function of a string Along with the observed standard two-particle correlation function: where one can introduce the string two-particle correlation function, Λ(η 1 , η 2 ), characterizing correlation between particles, produced from the one string: The Λ(η 1 , η 2 ) haracterizes the string decay properties (z − η correspondence) [X. Artru,Phys.Rept.97(1983) In this model we have the following connection: Nucl.Phys.A939(2015)21 ]. (Note that one often looses the constant part ω N / N of C 2 , using di-hadron correlation approach.) At midrapidities, implying uniform rapidity distribution: and the correlation functions depends only on a difference of rapidities: η sep = η 1 − η 2 = ∆η We suppose that the string correlation function Λ(∆η) → 0, when ∆η η corr , where the η corr is the correlation length.
where ∆η = η F −η B = η sep is a distance between the centers of the forward an backward observation windows. For a single string we have So in Σ(n F , n B ) we have the cancelation of the contributions from the fluctuation of the number of strings, ω N , and it becames strongly intensive: In general case the strongly intensive variable for a single string is defined similarly to Σ(n F , n B ) by It depends only on properties of a single string. So in the model with independent identical strings for symmetric reaction and small symmetric observation windows we found for Σ(n F , n B ): We see that really the Σ(η sep ) is strongly intensive quantity. It does not depend on N and ω N . Properties of Σ in model with independent identical strings The Σ(0) = 1 and increases with the gap between windows, η sep , because the Λ(η sep ) decrease with η sep , as the correlations in string go off with increase of η sep .
The rate of the Σ(η sep ) growth with η sep is proportional to the width of the observation window δη and µ 0 --the multiplicity produced from one string.
The model predicts saturation of the Σ(η sep ) on the level at large η sep , as Λ(η sep ) → 0 at the η sep η corr , where the η corr is a string correlation length. The parametrization for the pair correlation function Λ(η, φ) of a single string (reflecting the Schwinger mechanism of a string decay, was suggested in [V.V.,Nucl.Phys.A939 (2015)21]: This formula has the nearside peak, characterizing by parameters Λ 1 , η 1 and ϕ 1 , and the awayside ridge-like structure, characterizing by parameters Λ 2 , η 2 , η 0 and ϕ 2 (two wide overlapping hills shifted by ±η 0 in rapidity, η 0the mean length of a string decay segment). We imply that in formula (20) |ϕ| ≤ π .
If |ϕ| > π, then we use the replacement ϕ → ϕ + 2πk, so that (21) was fulfilled. With such completions the Λ(η, φ) meets the following properties is the e-by-e scaled variance of the number of strings, µ 0 is the average rapidity density of the charged particles from one string, i = 1 corresponds to the nearside and i = 2 to the awayside contributions, η 0 is the mean length of a string decay segment. Then we find Λ(∆η) integrating over azimuth: The obtained dependencies in this fugure for three initial energies are well approximated by the exponent: with the parameters presented in the This explains the general nature of the compensation for neighbor windows.   In this model we found that where we have introduce the Σ N i (n F i , n B i ) for i-th cell with N i strings by anology with (1) : If else all M cells are equivalent, P i (N i ) = P(N i ), then Using n = M n 1 and n (k) = M P(k) n 1 k it can be presented also as where k is a number of strings fused in a given sell and n (k) is a mean number of particles produced from all sells with k fused strings. α k =1. The same result was obtained in the model with two types of string in [E.V.Andronov,Theor.Math.Phys.185(2015)1383 ] for the long-range part of Σ(n F , n B ), when at ∆η η corr we have The same value of Σ(n F , n B ) in AA collisions, as in pp, if we suppose the formation of the same strings in AA and pp collisions. Because the Σ(n F , n B ) does not depends on the mean value, N , and the event-by-event fluctuations, ω N , in the number of strings. It depends only on string properties.
If we suppose the formation of new strings in AA collisions (and may be in central pp collisions at high energy) with some new characteristics, compared to pp collisions, due to e.g. string fusion processes, then for a source with k fused strings For these fused strings we expect, basing on the string decay picture [V.V., Baldin ISHEPP XIX v.1(2008)276; arXiv:0812.0604]: 1) larger multiplicity from one string, µ Both factors lead to the steeper increase of Σ k (η sep ) with η sep in the case of AA collisions, compared to pp.
In reality -a mixture of fused and single strings: Unfortunately in this case through the weighting factors α k = n (k) / n the observable Σ(n F , n B ) becomes dependent on collision conditions and, strictly speaking, can not be considered any more as strongly intensive.  (with E.Andronov) In case of zero net-charges in both windows, which is a very good approximation for mid-rapidity region at LHC collision energies we have: For symmetric reaction and small windows:   • The string model enables to understand the main features of the behavior of the strongly intensive observable Σ(n F , n B ). In particular the dependencies of this variable on the width of observation windows and the rapidity gap between them were found and its connection with the string two-particle correlation function was established.
• 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.
• In the case when the string fusion processes are taken into account and a formation of strings of a few different types takes place, it is shown, using a lattice in transverse plane, that this observable is equal to a weighted average of its values for different string types. Unfortunately in this case through the weighting factors the observable Σ(n F , n B ) starts to depend on collision conditions. In this model we have the following connection: C 2 (η 1 , η 2 ) = ω N +Λ(η 1 ,η 2 ) N [V.V.,Nucl.Phys.A939 (2015)21 ]. (Note that one often looses the constant part ω N / N of C 2 , obtaining C 2 by di-hadron correlation approach.) At midrapidities, implying uniform rapidity distribution: the correlation functions depends only on a difference of rapidities: Note that we use the two-particle correlation functions integrated over azimuth: C 2 (η sep ) = 1 π π 0 C 2 (η sep , φ sep ) dφ sep , Λ(η sep ) = 1 π π 0 Λ(η sep , φ sep ) dφ sep .