Using a strongly intense observable to study the formation of quark-gluon string clusters in pp collisions at LHC energies

Within the framework of the model with quark-gluon strings (color flux tubes) as sources, the properties of the strongly intense variable Σ, which characterizes the correlations between the number of particles in two observation windows separated in rapidity, are studied. It is shown that, in pp collisions at LHC energies, string fusion effects leading to the formation of string clusters have a significant effect on the behavior of this observable. The experimentally observed changes in this variable with the initial energy and centrality of the pp collision can only be explained by taking into account the formation of string clusters consisting of an increasing number of merged strings. It is demonstrated that the study of the behavior of the Σ observable as a function of the rapidity distance between the centers of the observation windows and the width of these windows with variable experimental conditions using different energies and centralities of pp-collisions makes it possible to extract the parameters of string clusters from the experimental data.


Advanced Observables
The string fusion processes under consideration affects both LRC and SRC. The LRC is sensitive to fluctuations in both quantity and type of sources. The SRC is sensitive to the properties of a single source (string) and its modification in a process of string fusion into string clusters.
Unfortunately for traditional observables the n F -n B correlation is strongly influenced by the "volume" fluctuations.
We can suppress the influence of these trivial "volume" fluctuations compared to the contribution of string fusion processes: 1) for LRC going from traditional extensive variables n F and n B to new intensive ones, 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)]).
2) for SRC going from b nn to more sophisticated correlation observables, e.g. to the strongly intensive observable Σ(n 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:

Σ in the model with independent identical strings
The fundamental characteristics of a string: one-and two-particle rapidity distributions from a single string decay: λ(η) = µ 0 , λ 2 (η 1 , η 2 ) = λ 2 (η 1 − η 2 ) = λ 2 (∆η) Λ(∆η) -two-particle correlation function of a string: δη -the width of the observation windows (below we suppose δη η corr ), ∆η = η sep -the distance between the observation windows. Vechernin V 2018  Properties of Σ in model with independent identical strings -We see that in the model with identical strings the Σ(∆η) is a really strongly intensive quantity. It does not depend nor on the mean number of strings N , nor on their event-by-event fluctuations ω N ≡ D N / N . It depends ONLY on string parameters: µ 0 and Λ(∆η).
-The rate of the Σ(∆η) growth with ∆η is proportional to the width of the observation window δη and µ 0 -the multiplicity produced from one string.

The model with identical strings
The ALICE data on b nn in pp ALICE collab., JHEP 05(2015) The string parameters occur dependent on initial energy (!?) The hint on the increase of the string cluster contribution to Σ(n F , n B ) with collision energy in pp collisions at LHC energies.
ACHT-2021 (21-23 April 2021) V. Vechernin 10 / 29 The model with identical strings Comparing the Σ(n F , n B ) with preliminary ALICE data In the model with string fusion on transverse grid we find Theor.Math.Phys. 200(2019)1094 ]: where k is a degree of string overlapping and n (k) is a mean number of particles produced from areas with such overlapping. α k =1. Here Σ k (µ F , µ B ) is the variable Σ for the cluster formed by k strings: where µ (k) 0 and Λ k (∆η) are the corresponding parameters of the string cluster.
Λ k (∆η) = Λ 1) larger multiplicity from one string, µ This corresponds to the analysis of the net-charge fluctuations in the framework of the string model for pp and AA collisions [A.Titov, V.V., PoS(Baldin ISHEPP XXI) 047(2012)].
Both factors lead to the steeper increase of Σ k (∆η) with ∆η and its saturation at a higher level That is in accordance with the energy dependence obtained above for Σ(n F , n B ) from the ALICE pp data.
for the case without string fusion in a given transverse cell.
(In last case Σ(n F , n B ) = Σ 1 (µ F , µ B ) and does not depends on α k .) The values of the parameters Λ (1) 0 = 0.8 and η (1) corr = 2.7 were chosen so that to obtain a correspondence with the values of the Σ(n F , n B ) obtained in [Vechernin V 2018 Eur.Phys.J.:Web of Conf. 191 04011].
Note that in that paper the Σ(n F , n B ) was calculated on the base of the string pair correlation function, Λ(∆η), extracted in [V.Vechernin, Nucl.Phys.A939(2015)  -Monte Carlo simulations of string configurations and calculation of weighting factors α k as a function of centrality and initial energy of pp collision.
where the m (k) is the mean number of clusters with k fused strings, which we take from our MC simulations of the string configurations.
-Calculation the Σ(n F , n B ) for different centralities of pp collision at few LHC energies using the relation (5).  The model with independent identical strings -In this version of the model the variable Σ(n F , n B ) depends only on the individual characteristics of a string and is independent of both the mean number of strings and its fluctuation, which reflects its strongly intensive character.
-So the studies of this observable enable to extract from the experimental data these fundamental characteristics of an individual string -a mean number of particles per unit of rapidity, µ 0 , and the pair correlation function, Λ(∆η, ∆φ), for particles produced from a fragmentation of a single string. -However in this version of the model the string parameters occur dependent on collision energy. This fact can be considered as a signal that with increasing of the initial energy of a pp collision due to the string fusion the formation of the sources with new properties] -the string clusters -takes place The model with string fusion and string clusters formation -In this case the observable Σ(n F , n B ) is equal to a weighted average of its values for different string clusters, Σ k (µ F , µ B ), with weight factors, α k , which are proportional to the mean number of the particles, produced from all clusters formed by the k fused strings.
-The Σ(n F , n B ), through these weight factors, α k becomes dependent on collision conditions -its energy and centrality.
-Analyzing these dependencies of the Σ(n F , n B ) we can extract from the experimental data the information on the individual characteristics of the string clusters -the multiplicity density, µ (k) 0 , and the pair correlation function, Λ k (∆η), for particles, produced from a decay of a given cluster. -In the framework of this approach it was shown that the overall increase of the Σ(n F , n B ) in pp collisions with collision energy and centrality can be explained by the formation of string clusters with new properties. Fitting the parameters of the initial string distribution in the impact parameter plane of pp collisions Table: The non-diffractive cross section, the multiplicity density at mid-rapidity and the mean number of initial strings in pp collisions at different initial energies.
If |ϕ| > π, then we use the replacement ϕ → ϕ + 2πk, so that (7) 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. [V.V.,Nucl.Phys.A939(2015) The string correlation function Λ(∆η) Then we find Λ(∆η) integrating over azimuth: The string correlation function Λ(∆η) The obtained dependencies in this fugure for three initial energies are well approximated by the exponent: with the parameters presented in the