Stability of Gluonic Systems with Multiple Soft Interactions

Rahul Kumar Thakur1, Bhupendra Nath Tiwari2,3 and Rahul Nigam1

1 BITS Pilani, Hyderabad Campus, Hyderabad, India.2 INFN-Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044 Frascati,

Rome, Italy.3 University of Information Science and Technology, “St. Paul theApostle”, Partizanska Str. bb 6000 Ohrid, Republic of Macedonia.

Abstract

In this paper, we investigate the stability properties of soft glu-ons in SIBYLL 2.1 with reference to its original version 1.7 thatcorresponds to hadronic hard interactions. In order to investigatethe stability structures, we classify the regions of the gluon densityfluctuations in its double leading logarithmic approximation and itsequivalent description as the fractional power law. In the parameterspace of initial transverse momentum Q and QCD renormalizationscale L that correspond to extensive air showers of cosmic rays, wehave categorized the surface of parameters over which the proton isstable. We further discuss the nature of local and global correlationsand stability properties where the concerning statistical basis yields astable system or undergoes a geometric phase transition. Finally wegive a phenomenological understanding towards the stability of softinteractions, Pomeron particle productions in minijet model, stringfragmentation and verify our result corresponding to the experiments- CDF, P238, UAS, GEUS and UA4 collaborations.

Keywords: Extensive Air Showers, Multiple Soft Interactions, Cosmic Rays,Gluon Density Fluctuations, Astroparticle Physics

1 Introduction

Cosmic rays interactions when viewed as high energy fixed target collisions,involve heavy particles with low intensity. Such interactions are exploredindirectly through Extensive Air Showers (EAS) Hiroaki et al.(2017). Inthe atmosphere, cosmic ray interactions are considered as the high energy

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fixed collisions involving various heavy particles. Because of their low inten-sity, cosmic rays with energies above 1015 eV can only be studied indirectlythrough the EAS as they are produced in the atmosphere. The analysis ofEAS relies on air shower Monte Carlo simulations which are used hadronicinteraction models. At higher energies, where the cosmic ray energy is be-yond the reach of man-made accelerators, hadronic interaction propertieshave to be extrapolated. The difficulties in the extrapolation are augmentedby the fact that, while the forward region contains most of the acceleratormeasurements are made in the central region. On the other, the explorationof the cosmic ray energy spectrum and its compositions for energies greaterthan about 1014eV are of prime interest, as well. An indirect analysis of thedata from EAS is interpreted in terms of the above mentioned spectrum andcomposition solely through the use of accelerator data.

Analysis of such air showers is performed via Monte Carlo simulationsusing an effective model of hadronic interactions, see Wang et al.(1991) to-wards multiple jet production in the light of pp, pA, and AA collisions. Inhigh energy phenomena, the known properties of hadronic interactions whichhave emerged via existing accelerators have to be extrapolated to ultra highenergies (≥ 1 TeV). For such cases, most of their energetics is contained inthe forward region as far as shower developments are concerned. Accelera-tor measurements of the interactions happening in the central region includemultiple soft interactions, parton density fluctuations, diffraction dissocia-tion etc. These measurements exhibit a better agreement with fixed targetcollider data Alekhin et al.(2018).

With the above motivations, we carry out an intrinsic analysis towards theunderstanding of the interactions happening in the forward region in the limitof SIBYLL 2.1. Specifically, we focus on fixed target hadronic interactions,collider experiments, dual parton model (DPM) Capella et al.(1994) and sub-gluonic interactions. The cosmic ray interaction event generator SIBYLL iswidely used in extensive air shower simulations. Hereby, we study the under-lying statistical fluctuation properties through the gluon density fluctuationsin the light of extensive air showers of cosmic rays. Our analysis follows fromthe minijet model involving hard interactions using a steeply arising gluondensity as a function of the model parameters, viz. the initial transverse mo-mentum Q of the colliding particle and QCD renormalization scale L Muta etal.(2009). To understand the hadronic interactions, we consider parametricdistribution where the hadrons-nucleus interactions are analyzed via Glauberscattering theory Glauber et al. (1970) and nucleus-nucleus interactions viasemisuperposition model Engel et al.(1992). Hereby, the string fragmentationinvolving quark-antiquark and quark-diquark pairs in the DPM picture arisevia air shower simulations in the light of cosmic rays Alekhin et al.(2018).

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In this paper, we concentrate on the stability analysis of such interactionsfor a varied range of energy scales. Note that the neutral pion equation arisesabove 1019 eV, which explains the small lifetime of these pions. It is worthmentioning that our model based on SIBYLL 2.1 is well suited for under-standing proton-proton scattering, parton structure functions, multiplicityfluctuations of charged particles in high energy diffractive events and softinteractions as mediated by an event generator Engel et al. (1999). Withthis perspective, our description of soft interactions and diffraction dissocia-tion is intrinsically contained in parametric fluctuations of the gluon density.Hereby, we have performed an analysis of gluon density fluctuation in orderto understand hadron-hadron/nucleus interactions at high energies, partic-ularly in the forward region. Namely, we primarily look for the parameterspace that lead to stable secondary proton which is obtained from the studyof the stability of gluon density under variations of its model parameters.

Following the minijet setup of the phase space containing QCD improvedparton whose transverse momentum satisfies the energy dependent relations,we explore the stability of quark-quark and quark-diquark pairs connectedvia an effective string. Here the sampling of the momentum fraction is donevia fractional energy distribution of the quark. In this concern, the fractionalenergy distribution of diquark is obtained via the corresponding fractionalenergy distribution of quark within effective quark mass. To consider sta-bility property of multiple soft interactions, one uses Regge theory with adependence of the impact parameter in contrast to the hard interactions,where the Good-Walker model (Good et al.1960,Fletcher et al.1994)plays animportant role as a two channel eikonal model.

Interestingly, our model is well applicable for low and high mass diffrac-tion dissociation despite the fact that they are not well understood in theconventional gauge theory of hadronic interactions and collider physics. Mul-tiple soft interactions as well as new partons yield large multiplicity at highenergy, whereby diffraction dissociation phenomena offer an improved under-standing of multiplicity distribution of such particles. Herewith, we offer anintrinsic analysis of hadronic interactions towards the understanding of theirforward region where most of the particles are expected to be found in thelight of the extensive air shower developments.

In the above description of the gluon density function as a function ofthe fractional energy and transverse momentum mediated by small couplingconstant satisfying a geometric saturation relation, we explore an ensemble ofQCD configurations in order to determine the statistical stability of a protonin the setup of a minijet model, which holds beyond the conventional collinearfactorization approximation Lipatov et al.(1997). As far as phenomenolog-ical events are concerned, the output of our analysis matches well the with

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collisions mediated via soft interactions and this helps in the understandingof deviations of theoretical predictions with their corresponding data coun-terparts. In particular, the behavior shown by transverse momentum as afunction of energy varies in accordance with the predictions of SIBYLL 2.1.

The effect of gluon density fluctuations on physical observables such asthe multiplicity and pseudo-rapidity distribution is described as follows: Boththe fixed target experiments and collider counterparts provide valuable in-formation in the modeling of hadronic interactions. Namely, the fixed targetexperiments yield data in the forward region of interactions that are mostrelevant towards the study of cosmic rays. However, the energies that arerelatively lower than the laboratory frame energy Elab are of the order ofseveral hundred GeV. Namely, in collider experiments, one can probe suchhigh energies with the laboratory frame energy Elab ∼ 106GeV . Followingthe same, most of the information is collected for the central region of colli-sions. Nonetheless, there are experiments such as H1 and ZEUS Breitweg etal. (2002) that are capable of detecting the forward region events. Namely,the experiments such as LHCf and TOTEM Ahn et al.(2009) in the LHC atCERN are supposed to collect data in the forward region at energy scalesthat are equivalent to cosmic rays, viz. Elab ∼ 108GeV .

These experiments produce more particles with an augmented distribu-tion in momentum space that are evident when comparing the associatedcentral region data. In nutshell, both the versions SIBYLL 2.1 and SIBYLL1.7 have offered a good fit to the rapidity and Feynman distribution for chargeparticles for pp-collisions in fixed target experiments such as NA49 Gornayaet al.(2018). It is worth mentioning that the changes made in SIBYLL 2.1 arevery evident in the central region, namely, one observes a distinct change inthe pseudorapidity and the overall multiplicity distribution of charged parti-cles Ahn et al.(2009). As a result, in SIBYLL 2.1, the air showers are quicklydeveloped with smaller shower maximum and larger muon number than itsformer counterpart. A comparative analysis of SIBYLL 1.7 and SIBYLL 2.1is given in Section 2.1 below.

In the light of cosmic rays air showers, the transverse momentum andrenormalization scale on the multiplicity and pseudo-rapidity distributionplay an important role in selecting the forward region of interactions. In thisconcern, we have selected a forward region of interactions by properly choos-ing the range of the Mandelstam variables as the energy of the constituentquark and diquark pairs. At higher energies, where the energy of cosmic raysis beyond the reach of a man-made accelerator, the hadronic interactions areunderstood by extrapolation techniques Ahn et al.(2009). In such extrapola-tions, the forward region contains most of energetic particles that support theformation of shower developments of cosmic rays. However, due to the string

4

fragmentation involving quarks and diquarks, the difficulties are increasedwith the fact that the most of accelerator measurements are performed inthe central region Ahn et al.(2009), whereby one obtains a saturated gluondensity profile in its the double leading-logarithmic approximation. In thispaper, we study fluctuations of the corresponding gluon density in the spaceof model parameters that leads to stable secondary protons.

In the light of the strong interactions, there exist recent investigationstowards the QCD phase structures, their decoding and particle productionat high energy Andronic et al. (2018). This study is based on the latticeMonte Carlo simulations of the phases of QCD. At high temperature, it isanticipated that there is a phase change from a confined hadronic matter toa deconfined quark-gluon-plasma, where the quarks and gluons can traveldistances that critically exceed the size of hadrons (Andronic et al.2018, An-dronic et al.2018). Following the same, it is widely believed that quantumchromdynamics should undergo a phase transition at a finite temperatureand/ or baryon density (Kalashnikov et al.1979, Bellwied et al.1994). Fur-thermore, the chiral symmetry should be restored and the quarks should bedeconfined at a sufficiently high temperature with a definite baryon chemicalpotential (Stephanovet al.2006,Meyeret al1996,Chenget al.2006). Perspec-tives phase transitions (or more generally, the QCD thermodynamics) areessential tools for an understanding of the underlying physics of early uni-verse, as well as, the core of neutron stars and associated laboratory experi-ments at relativistic heavy ion collisions (Rapp et al.2002,Basset al.1999, Heet al.2007). With this motivation, we explore the stability structures of glu-onic systems with multiple soft interactions in the light of background colorfluctuations in minijet models. Namely, we analyze gluon density fluctua-tions in double leading logarithmic approximation to understand extensiveair showers of cosmic rays. Prospective study of its next version and a com-parative analysis of LHC data are left open for future research.

The aim of this paper is to offer a statistical understanding of the stabilityof hadronic interactions and their associated phase transition phenomenasatisfying double leading logarithmic approximation in terms of the gluondensity function with an identical Bjorken scaling and transverse momentumof the scattered partons. In this concern, one has a steeply increasing gluondensity at fractional energy as exhibited in the results of HERA (Adloffet al.1997, Breitweg et al.1997). Note that parton densities in the limit ofSIBYLL 2.1 discussed in GRV data (Gluck et al.1995, Gluck et al.1998) showsthat gluon density as a function of fractional energy x scales as x−(1+δ) where0.3 ≤ δ ≤ 0.4. In this regard, we provide a brief overview of the gluon densityfunction in section 2 and associated fluctuation theory perspective section 3.In the sequel, in section 4, we provide fluctuation theory analysis of the gluon

5

density fluctuations with their qualitative discussion in section 5. In section6, we give the conclusion of our analysis with perspective direction for thestudy of random observation samples.

2 Gluon density function: an overview

In this paper, we study statistical fluctuations of gluon density in the limitof double leading logarithmic approximation that is well suited for study ofhadronic collision and in particular the soft interactions.

2.1 SIBYLL event generators

First of all, the most important changes in the version SIBYLL 2.1 are that itprovides the description of soft interactions and diffraction dissociation, whileits former version SIBYLL 1.7 deals only with hard interactions. Namely, inorder to allow multiple soft interactions as in SIBYLL 2.1 , the eikonal forthe soft interactions is described by using Regge theory, whereas in SIBYLL1.7 version the eikonal for the hadronic interactions was energy independentand it had the same impact parameter dependence as used for hard interac-tions. Hereby, the distribution of the secondary particle multiplicity, energyvariation of multiplicity, the distribution of secondary particles at high x val-ues and others depending in the soft interactions will change. Namely, inversion SIBYLL 1.7, the cross section for diffraction dissociation is param-eterized independently of the eikonal of the model, that is, a two-channeleikonal model based on Good-Walker model, see Good et al.(1960) for theassociated fundamentals of SIBYLL 1.7 and SIBYLL 2.1.

Gluons are the glue-like particles that bind the quarks within the baryonsand mesons Mutta et al.(2009).In Quantum Chromodynamics (QCD) Mutaet al.(2009) high energies fluctuation effects arise when the Pomeron loopequations are used to describe the dipole evolution with an increasing ra-pidity Hiroaki et al.(2017) as an effective field theory. In the kinematicalrange, the description of such an QCD-improved dynamics of colliding par-ticle which will be probed in the future electron hadron colliders is still anopen question. However, the corresponding phenomenological investigationsindicate that the gluon number fluctuations are related to discreteness in theQCD evolution that are negligible at the energy scale of HERA Sarkar etal.(2018). Here, the magnitude of such effects play an important role in thenext generation colliders Amaral et al.(2014). In the realm of QCD, it isknown that the gluon gluon interactions dominate the production of bottomquarks at hadron collider energies. On other hand, the gluon-quark interac-

6

tions interpolate inclusively to prompt photon production at a large trans-verse momentum in pp collisions at fixed-target energies Berger et al.(1992)

The underline uncertainties of the gluon density as extracted from theglobal parton model is large enough in the kinematical range when theBjorken variable takes a small value at the scale of hard interactions Goncalveset al. (2016).The early measurements have revealed a compelling evidencefor the existence of a new form of nuclear matter at extremely high densityand temperature. This yields a medium in which the predictions of QCD canbe tested, whereby new phenomena can be explored under conditions wherethe relevant degrees of freedom over a given nuclear volumes are expected toarise from the interactions of quarks and gluons rather than that of hadrons.This lies in the realm of the quark gluon plasma, where the predicted stateof matter whose existence and properties are under exploration at the RHICexperiments Arsene et al. (1978)

2.2 Gluonic matter formation

In the setup of SIBYLL 2.1, the parton distribution functions (PDF) ofthe proton are essential in order to make theoretical predictions and po-tentially obtaining developments towards the physical understanding of theexperimental results arising at high energy colliders. An accurate determi-nation of such PDFs and the undermining uncertainties lying in the globalanalysis are therefore crucial. There have been various considerations inthis direction by several groups of researchers,(Alekhinet al.2010,Sjostrandetal.2006,Pumplinet al.2009,Aaronet al.2010,Khozeet al.2018) In this paper,we offer theoretical predictions towards the stability of gluon density func-tion under the variation of its model parameters and compare the resultswith the existing experiments. In addition, we offer the prediction towardsthe stability of gluonic matter formation.

As far as the deep-inelastic scatterings (DIS) are concerned, the hardproton-proton collisions or proton-antiproton collisions at high-energies hap-pen via the scattering of partonic constituents of hadrons. To predict the rateof these processes, a set of universal parton distribution functions would berequired. Such distributions are well determined by a global fit to the existingresults via DIS and associated hard-scattering data Martin et al.(2009). Atultrahigh energies, the cosmic ray interactions in the atmosphere can be ex-plored with fixed target collisions involving heavy particles. Because of theirlow intensity, cosmic rays with energies above 1015 eV can solely be studiedthrough the extensive air showers (EAS) which analyses such interactions inthe atmosphere Ahn et al.(2009).

Herewith, one of the central predictions of QCD is the transition from

7

the confined phase that is chirally broken to the deconfined phase which ischirally symmetric state of quasi-free quarks and gluons. The above setupis termed as quark-gluon plasma (QGP) which has been explored via heavyion colliders (Heinzet al.(2000).Goncalveset al.2002) While the high energyhadronic interactions studied at the LHC involve QCD effects that are yetto be well understood as they remain largely unexplored experimentally. Itwould be interesting to push further the above study of soft interactions atultrahigh energies in order to detect the cosmic rays in the atmosphere.

2.3 Ultra high energy colliders

At this juncture, by extrapolating Monte Carlo simulations to ulta high en-ergies, one often resorts to understand such phenomenological models (Sjos-trandet al.2006, Bahr et al.2007, Ranft et al.1995, Roesler et al.2001, Engel etal.2001, Drescher et al.2001, Ostapchenkoet al.2001, Kalmykovet al.1979, Sci-uttoet al.2001,Engelet al.1992, Fletcheret al.1992). These studies are largelybased on fits to associated lower energy data. For a given QCD renormal-ization scale, our interest lies on contributions emerging from jets with agiven initial value of the relative transverse momentum in the light of mini-jet or semihard jet models. To understand the internal structure of nucleons,the nucleon-electromagnetic form factors are used in the existing modelsPankajet al.(2015) Fundamentally, the internal structure of the nucleon in-volve the electromagnetic Dirac and Pauli form factors that are equivalentto the electric and magnetic dipoles. Such form factors are examined at thelevel of experiments and phenomenology Arringtinet al.(2007).

In particular, in order to test the stability of gluonic matter formation,calculus based tools are used to find the maxima, minima and saddle point ofthe gluon density function. Namely, we carry out the analysis of the densityprofile function in the double leading logarithmic approximation in order tounderstand the hadron-hadron interactions. The fluctuations are examinedin terms of the model parameters. Thereby, we are interested in finding theregion of the parameter space where a proton is stable. In the next section,we discuss such regions in detail with the associated analysis of the formationof metastable particles like Pomerons, Reggions and other particles that maybe detectable with a small probability.

It is worth mentioning that the SIBYLL 2.1 retains the DPM picture ofhadronic interactions. Namely, in the DPM picture, a nucleon is composed ofquarks in particular as a color triplet, and diquarks as a qq-color antitriplet.Herewith, the soft gluons are exchanged via an interaction and the color fieldsget reorganized as the interactions proceed. In such a model, the projectilequark or diquark combines with the targeted diquark or quark. Such a pair

8

of quarks and diquarks is connected by a pair of strings. In due course, eachstring fragments separately via the Lund’s string fragmentation model Engelet al. (1999) Scattering processes at high energy corresponding to the hadroncolliders are classified as either the hard or soft interactions. Hereby, it followsthat the QCD plays a vital role in understanding the underlying theory ofsuch processes. The methodology and level of the understanding vary in boththe cases. For hard processes such as the Higgs boson or high transversemomentum jet production, the rates and associated stability properties arestudied by using perturbation theory.

3 Review of the proposed model

In this section, we offer an overview of soft interaction in order to discusstheir stability properties. We are interested in studying hadron-hadron col-lisions by invoking the role of gluon density and associated QCD improvedparton models Arringtin et al.(2007) form factors Heikki et al.(2016) andjet quinching Amaral et al.(2014) We study optimization of hard/ soft in-teractions by using a steeply rising gluon density function corresponding toimpact parameter dependent soft profile function of a proton. Given the softprofile function for a proton-proton collision as in Hiroaki et al.(2017) themodel parameters are the QCD renormalization scale L and initial transversemomentum of the colliding particle Q. Here, the value of parallel componentplays the role of the soft-hard function and the transverse component deter-mining the effective radius of the proton.

Notice that the choice of the parameter L and Q yields a hard profilein the limit of SIBYLL 1.7. Here, the parameters {L,Q} represent a twodimensional surface in the ensemble space of QCD-improved models of par-tons. On the other hand, the choice of the parameter L,Q depends solelyon the geometric saturation condition of the colliding profile function of thequark-quark and quark-diquark pairs. Classically, the longitudinal compo-nent of the transverse momentum of the colliding parton satisfies a sharpclassification boundary between the energy independent and energy depen-dent generators as far as the elastic and inelastic collisions are concerned inthe light of SIBYLL 2.1 Ahn et al.(2009)

3.1 QCD-improved parton model

Importantly, the cosmic ray interaction event generator plays an importantrole Hiroaki et al.(2017) In this case, we aim to study stability of an interact-ing system with multiple soft interactions via the fluctuations of the gluon

9

density in the limit of steeply rising double leading logarithmic approxima-tion. We provide saddle point analysis of soft/ hard interactions and analyzethe undermining statistical phase transitions, see Aurenche et al.(1992). forthe data concerning the centre of mass of transverse component. Based onthe dual parton model Aurenche et al.(1992) we classify stability property ofhadronic interactions via proton-proton collision with a fixed energy scale aswell as an energy dependent scale satisfying geometric saturation conditionArringtin et al.(2007) in terms of the strong coupling constant

αs(p2T )

p2Txg(x, p2T ) ≤ πR2

p, (1)

where Rp is the effective radius of the proton in the transverse space, pT isthe transverse momentum and x is the fractional energy of quarks in DPMpicture. Physically, our motivations of the hadronic matter stability analysisfollow from the elastic and inelastic cross sections for p-p and p-p interac-tions Hiroaki et al.(2017). Hereby, the hard interaction happens at a pointthat correspond to approximately corresponds to a position where we have aconstant value of the transverse momentum of the QCD improved partons.

Given the above qualification of soft interactions, we have a soft profilefunction corresponding to energy dependent minijet production via steeplyrising gluon density, whereby we explore its fluctuations in the space of theparameters L and Q. As in Hiroaki et al.(2017). the hadron-hadron colli-sion in the limit of double leading approximation going beyond the collinearfactorization approximation are expected to correspond to a Gaussian distri-bution as a function of {L,Q} which is the effective radius of a proton in thetransverse space. Namely, for a proton-proton collision Hiroaki et al.(2017)mediated via the above geometric saturation condition in the limit of minijetmodel, with the minimum transverse momentum q, the corresponding gluondensity of soft interactions Ahn et al.(2009) is given by

xg(x, q2) ∼ exp

[48

11− 23nf

loglog q2

L2

log Q2

L2

log1

x

]1/2(2)

Subsequently with the above limiting gluon density function 2, we findits maxima/ minima of in the space of its parameters {L,Q} at a fixedb fractionation energy and effective radius of the proton in the transversemomentum with its dependence on the Bjorken scaling Q2 = p2T and thecentre of mass energy square s. Herewith, we study the fluctuations of thegluon density concerning the stability of proton or other baryons, see Andreaset al.(2010).and A.S. Kronfeld Nora et al.(2004) provides a comprehensivereport on quantum chromodynamics and associated heavy quark physics.

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One may further consider optimization of multiparticle profile in the light oftwo-component dual parton model, see Aurenche et al.(1992)

Let us focus on the pp collisions with its corresponding profile function asa vector valued function in the st-plane. Thus, the respective optimizationof the boundary of soft intersection is obtained via the maxima/minima ofthe associated gluon density function in the space of its parameter L andQ. Associated physical properties arises via the Bremsstrahlung, see Claudeet al.(2012). In the light of classically field theory, variations of the actionabout its fixed points yields Euler Langrage equations. Here, the variationsare performed with respect to all possible fields about fixed points of theundermining action integral of the theory, see in text in QFT Claude etal.(2012). chapter 2 and also in QFT Ashok et al. (2008) Furthermore, ourfluctuation theory based study can incorporate effects of the optimal softmagnetic field, as well.

3.2 Stability of fluctuating models

Given the gluon density function f(Q,L), its fluctuations are described throughthe embedding function reading as

f :M2 → R, (3)

In the above case of two fluctuation variables (Q,L) with the above em-bedding f(Q,L), it follows (Ruppeiner et al. (1995), Tiwari et al. (2011),Bellucci et al. (2010), Tiwari et al. (2011), Bellucci et al. (2013), Bellucci etal. (2013), Tiwari et al. (2018), Bellucci et al. (2018), Aman et al. (2006),Bellucci et al. (2010), Tiwari et al. (2011), (Weinhold Frank em et al. 1975);Ruppeiner et al. (1983), Ruppeiner et al. (1979), Ruppeiner et al. (1983),Millan et al. (2010), Simeoni et al. (2010), Widom et al. (1972)) that thecorrelation discontinuities give the nature of undermining phase transitionson the fluctuation surfaceM2. This arises when the system of the fluctuationmatrix as in Eqn.(4) vanishes as a function of the QCD parameters {Q,L}.

Herewith, from Eqn.(3), the stability analysis is carried around its criticalpoints arising as the roots of the flow equation fQ = 0 and fL = 0 byjointly evaluating the signs of one of the heat capacities {fQQ, fLL} and thefluctuation determinant ∆ := |H| as the first and second principal minors ofthe concerning fluctuation matrix

H :=∂2

∂xi∂xjf(Q,L) (4)

for the chosen parameter vector ~x := (Q,L) on the surface Σ of parameters{Q,L}. In the light of statistical mechanics, we wish to classify the nature

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of an ensemble of QCD-improved Pomeron configurations with their gluondensities as in Eqn.(2) having values of the parameters as

Σ := {(Q(1), L(1)), (Q(2), L(2)), (Q(3), L(3)), . . . , (Q(N), L(N))} (5)

where it corresponds to a (non)interacting statistical basis as we take N →∞.

In the above set up, it is worth mentioning that we have a stable configu-ration when both the first principal minor p1 := fQQ and the second principalminor p2 := ∆ take a positive value as defined in the next section. Noticefurther that the undermining system corresponds to an unstable statisticalsystem when p2 is positive and p1 is negative. It is worth mentioning that thesystem corresponds to a saddle point configuration when we have a negativevalue of p2 for any value of p1, under fluctuations of the parameters {Q,L}.

3.3 Fluctuations in collision data

Now we determine the optimum value of the transverse momentum whichwould correspond to soft profile function of proton. To be precise, this arisesas a type of Taylor series expansion of the gluon density as in Eqn.(2) withrespect to the parameters Q,L. It is worth mentioning that the centre ofmass component corresponding to the undermining pp collisions satisfies theRegge trajectory Hiroaki et al.(2017). Statistically, the parameters Q,L givethe variance of the gluon parametric density fluctuations that in the case ofthe collision data corresponds to a fuzzy area of approximate radius of theproton in transverse space.

The interaction area can be parametrized as a Gaussian fluctuation profilewith an energy dependent width. Following the fundamental central limittheorem of statistics Huang et al.(2009 ) it follows that every fluctuatingsample leads to a Gaussian profile in long term limit of interactions. In thisconcern, Ref. (Lucien et al.1986, Dmitryet al.2017)provides the proof of theclassical central limit theorem and its variants. Physically, the constantssuch as the impact parameter play an important role in understanding thestatistical behavior of a chosen pair of protons undergoing the collisions.

4 Gluon density fluctuations

In this section, following the fact that fixed target and collider experiments of-fer valuable guidance in modelling hadronic interactions Hiroaki et al.(2017),we provide fluctuation theory analysis of gluon density function. Note thatin the setup of Lund string fragmentation model, if the quark has a stable

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distribution then the diquark will have an unstable distribution in the limitof DPM picture.

Physically, fluctuations of an ensemble of gluons will continue in a dampmanner untill the remaining mass of the string becomes smaller then thethreshold mass of the quark/diquark pair mass. In other words the fluctua-tions stop when two final hadrons are formed. To discuss the above phenom-ena, one typically includes high multiplicity, increase in the mean transversemomentum, high transverse momentum jet and increase in the central rapid-ity density. The energy fractination plays an important role in high energyLipatov et al.(1997) whereby the minijet model based on QCD improvedparton configuration is expected to have an energy dependent transversemomentum cutoff Arringtin et al.(2007). In the limit of double leading ap-proximation, we have steeply rising gluon density profile as in Eqn.(2).

Further we study fluctuations of the gluon density as in Eqn.(6) as afunction of the initial transverse momentum Q and QCD renormalizationscale L. In order to do so, we may examine the corresponding fluctuationsof the gluon density Ahn et al.(2009) in the space of {Q,L} as the followingfunction

f(Q,L) = B loglog q2

L2

log Q2

L2

, (6)

where the coefficient B is related to the gluon density g(x) at a fraction ofenergy x and number of flavours nf by the relation

B = exp

(− (log(

xg

k))2

1

log 1x

11− 23nf

48

)(7)

4.1 Local fluctuations

To study fluctuations of the above gluon density, we need to compute thecorresponding flow components of gluon density f(Q,L). In this case, seethat the Q-flow component can be written as

df

dQ= −2B

Q

log L2

q2

(log Q2

L2 )2(8)

Similarly, it follows that the flow component corresponding to QCD renor-malization scale L reads as

df

dL=

2B

L

log q2

Q2

(log Q2

L2 )2(9)

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Hereby, the flow equations fQ = 0 and fL = 0 yield the value of L andQ as L = ±q and Q = ±q respectively. Thus, it would be interestingto examine the nature of statistical interactions near an equal value of theQCD parameters Q and L. To examine the nature of the stability of the gluondensity profile as in Eqn.(6), we need to compute the fluctuation capacitiesdefined as its second pure derivatives with respect to the system parameters{Q,L}. In this case, we see that the pure Q-capacity of the gluon densitysimplifies as

d2f

dQ2=

2B

Q2

log q2

L2

(log Q2

L2 )3

(4 + log

Q2

L2

)(10)

Therefore, at an equal value of the model parameters Q = L, we find thatthe fluctuation capacity fQQ diverges to a large positive value. Similarly, thecorresponding pure L-capacity is given as per following expression

d2f

dL2=

2B

L2

log q2

Q2

(log Q2

L2 )3

(4− log

Q2

L2

)(11)

In the above case as well, at an equal value of the model parameters Q = L,the fluctuation capacity fLL diverges to a large positive value.

4.2 Local correlation

The undermining local correlation following the above gluon density as de-picted in Eqn.(6) is given by its corresponding mixed derivative fQL as fol-lowing

d2f

dQdL= − 4B

QL

log q4

Q2L2

(log Q2

L2 )3(12)

Thus, at the value of the model parameters Q = q, for all values of q, L ∈ R,the local cross correlation fQL is given as

d2f

dQdL|Q=L = −4B

q2(log

q2

L2)−2 (13)

Later we study the equations at the critical values of the model parametersQ = q and L = q to find that the local cross correlation fQL diverges to alarge negative value.

4.3 Global fluctuations

In order to examine the global stability of the ensemble under fluctuationsof {Q,L}, we are required to compute the fluctuation discriminant ∆ :=

14

fQQfLL − (fQL)2. Substituting the above expressions for fluctuation capac-ities {fQQ, fLL} and cross correlation fQL, we find the following fluctuationdiscriminant:

∆(Q,L) = − 4B2

Q2L2(log(Q2

L2 ))4

(4 + log(

q2

Q2) log(

q2

L2

)(14)

Finally, at the value of Q = q, the determinant of the fluctuation matrixmodulates as

∆(Q,L) = − 16B2

Q2L2

(log(Q

2

L2 )

)4 (15)

We note that at the critical values of the model parameters Q = q andL = q, the discriminant ∆ undermining the global correlation diverges toa large negative value. Thus, as we approach the parametric fixed point(Q,L) = (|q|, |q|), we have an unstable configuration as both the fluctuationcapacity {fii | i = Q,L} and fluctuation determinant diverge.

4.4 Physical Perspectives

Note that when one of the parameter critical points Q2 = q2 is approachedwhile the QCD renormalization scale L remains fixed, we have a locally stableconfiguration as the respective heat capacities fQQ or fLL take a positivevalue. However, even in the limit of approaching the transverse momentumsquare q = ±Q while L2 remains fixed, the system remains unstable as thedeterminant ∆(Q,L diverges.

In the light of QCD phase transitions, we see that the correlation lengthdoes not diverge unless ∆ = 0. That is, the point {Q,L} in the parameterspace M2 satisfies

4 + logq2

Q2log

q2

L2= 0 (16)

Further, we have zero correlation length when the fluctuation determinant∆ → ∞. This happens also when we either have |Q| = ∞ or |L| = ∞ orthe function B(nf , x) = 0 as in Eqn(7). This would never happen in thisfluctuation theory model of Pomerons because the function f(Q,L) vanishesidentically in this limit.

On the other hand, there are no statistical correlations when the corre-lation area l2 ∼ ∆−2 = 0. This happens for the following limiting values ofthe model parameters: {Q,L} as Q = 0 or L = 0 or Q = ±L, or when thepoint {Q,L} in the space of model parameters satisfying

4 + log(q2

Q2) log(

q2

L2)→∞ (17)

15

It is evident that the Eqn.(17) holds whenever we have either q >> L orq >> Q. Later we discuss the statistical interpretations the above resultsconcerning the local and global correlations undermining an ensemble ofQCD-improved Pomeron configurations where the associated gluon densityis considered as a function of the parameters {Q,L}.

4.5 Statistical Interpretations

• In the light of the above analysis, the corresponding system is expectedto yield a locally stable statistical basis when either of the heat capac-ities {fQQ, fLL} has a positive sign. For q >> L, the capacity fQQhappens to be positive when the factors log Q2

L2 and (4 + log Q2

L2 ) havethe same sign. That is, for Q2 > L2, we have the following conditionQ2 > L2/e4. On the other hand, for Q2 < L2, the fluctuation capacityfQQ take a positive value for Q2 < L2/e4.

Therefore, the local instabilities are expected to be detected in theregion L2/e4 < Q2 < L2. Hereby, the test of stability needs furtherexaminations in this band of parameters {Q,L} as well as at theirextreme values Q = ±L, q = ±L, and q = ±Q. Similarly, if we choosethe QCD renormalization scale L as the first coordinate in the surfaceof the parameters as {(L,Q)|L,Q ∈ R}, for the choice of the transversemomentum q > Q, the capacity fLL happens to be positive when thefactors log Q2

L2 and (4− log Q2

L2 ) have the same sign. The analysis followsas above in the case of fQQ.

• The corresponding local correlation is defined as the mixed derivativeof f(Q,L) that reads as the ratio

fQL = − 4B

QL

log q4

Q2L2

(log Q2

L2 )3(18)

For Q > 0 and L > 0, note the local correlation fQL as defined aboveremains negative for either Q2L2 > q4 and Q > L or Q2L2 < q4 sinceq2 >> L2 since q2 >> L2. Physically, in this case, their exists a locallycontracting statistical system. On the other hand, forQ > 0, L > 0, wehave a locally expanding system when fQL takes a negative value, thatis, we have either Q2L2 < q4 and Q > L, or Q2L2 > q4 and Q < L.

• In this case, the phase transitions occur when the global correlationlength l → ∞. From Eqn.(15), we see that this happens precisely as

16

per the correlation area

l2 ∼ − 1

4B2

Q2L2(log Q2

L2 )4

(4 + log q2

L2 log q2

Q2 )(19)

• For the choice of Q = L, from the Eqn.(15) it follows that ∆(Q,Q)→∞. Thus, the undermining correlation length l ∼ ∆−1 → 0. Phys-ically, in the light of statistical mechanics, we may interpret that anensemble of QCD-improved Pomeron configurations with their gluondensities having an ensemble of values {(Q(i), L(i)) | i = 1, 2, . . .∞} ofthe QCD parameters {Q,L} correspond to a non interacting statisticalconfiguration.

5 Discussion of the results

In this section, we give an interpretation of the results and offer their quali-tative analysis in the light of existing literature.

Below, in the fig (1) we offer a qualitative behavior of the minimumtransverse momentum following the geometric saturation condition Hiroaket al.(2017). For different value of the QCD renormalization scale, we seethat it has a varied range of behavior namely for L = 0.0065, Q has a similarbehavior as SIBYLL 1.7 that uses constant parton density parameterizationwith energy independent transverse momentum cut off for proton-proton andproton-antiproton scattering cross section. For a higher value L = 0.065, wesee a larger flow of Q plotted with respect to the energy in logarithmic scalewith its discrete values as the following definition:

lnE(j)i =

1

2c2

(log(

qiLj

)(1− exp(− 1

log( qiLj

))

)2

, j = 1, 2, 3 (20)

Furthermore, for an increase value of the QCD renormalization scaleL = 0.65, the curve nearly becomes a straight line originating from theorigin of the space of transverse momentum and energy in the logarithmicscale. Hereby, from fig.(1), we see that our fluctuation theory analysis isindispensable for a larger value of renormalization scale L. In other words,the SIBYLL 1.7 is acceptable for a smaller QCD renormalization scale whichis physically not the case in order to have asymptotic freedom of the partons.

In contrast to the minijet based SIBYLL 2.1, viz. the figure [1] of Hiroaketal.(2017), where the minimum transverse momentum is plotted with respect

17

E(GeV) 1 10 30 53 200Q(GeV) 1.24 -13.12 -25.41 -33.88 -60.77E(GeV) 210 630 1000 1800Q(GeV) -61.98 -94.30 -111.16 -135.87

Table 1: A comparative analysis of the flow of the transverse momentum Qwith respect to energy E that are measures in GeV.

Figure 1: The initial momentum q plotted as a function of the energy Ein the logarithmic scale plotted on X-axis and q on Y -axis describing thenature of flow of the momentum by considering variations of the energies Efor different values of L. Here, L1 = 0.065 is depicted by the red solid line,L2 = 0.65 by the blue dashed line and L3 = 0.0065 by the black dash-dotline.

to the center of the mass energy, the minimum transverse momentum hasa vanishing energy cutoff value of 1.5 GeV and reaches the SIBYLL 1.7cut-off at 2.2 GeV. Here we note that the respective values of the minimumtransverse momentum values of the zero, 1.7 and 7 in GeV scale of the energyE. For a nonzero value of the energy, we see a parabolic behavior as the valueof E increases in the logarithmic scale.

In this concern, a comparative analysis of the flow of the transverse mo-mentum Q with respect to energy E is summarized in the Table 1.

18

Figure 2: The initial momentum Q plotted as a function of the energy Ein the logarithmic scale plotted on X-axis and Q on Y -axis describing thenature of the momentum flow by considering variations of the QCD re nor-malization scale L for different values of E. Here, the string fragmentationwith the energy E = 1 MeV is depicted by the cyan solid line, the diffractiondissociation pomeron with energy scale E = 10 MeV by the red dashed line,minijet production with energy scale E = 30 MeV by the magenta dottedline and the soft interactions with energy scale E = 10000 MeV by the bluepoint line.

In the fig.(2), we provide qualitative discussion of diffraction dissociationbased on the pomerons, soft interaction, string fragmentation and particleproduced in minijet model with the definition of the energy as

Q(i, j) = q − L(j) exp(c√

2 log(E(i))), (21)

where E reads as above the caption of fig.(2, 3) and L ∈ (0, 1). Namely,from fig.(2), we find that the string fragmentation arises with a highest trans-fer momentum q and soft interaction arise with a lowest transfer momentumq when they are varied with respect to QCD renormalization scale L. In thisanalysis, we observe that the pomeron have a larger transverse momentumcutoff in comparison to the minijet model based particle productions. In allthe above cases, we notice that the linear momentum cutoff almost behavedlinearly with respect to the QCD renormalization scale.

In the fig.(3), we offer a qualitative analysis of the transverse momentumcutoff when it is varied with respect to the QCD renormalization scale in or-der to compare with the existing QCD collaboration energy scale Hiroakietal.(2017). Hereby, from fig.(3), we see that UAS energy scale has a larger

19

Figure 3: The initial momentum Q plotted as a function of the energy E inthe logarithmic scale plotted on X-axis and q on Y -axis describing the natureof the momentum flow by considering variations of the energies E for differentvalues of L for various collaborations. Here, the UAS collaboration E =53MeV is depicted by the cyan solid line and its next energy E = 200MeVby the red dashed, GEUS collaboration energy scale E = 210MeV by theblack dotted line, P238 collaboration with its energy scale E = 630MeVby the green dash-dot line, and CDF collaboration with its energy scaleE = 18000MeV by the blue point line.

transverse momentum cutoff in comparison with the GEUS, P238 and CDFcollaboration scale a comparative analysis of the transverse momentum cut-off for the energy scale of these collaboration shows that Q generally takesa negative value which physically indicates that the undermining particle isproduced by absorbing a finite amount of energy. Namely, from fig.(3), wenotice that the absorption energy increase as we increase QCD renormaliza-tion scale.

In the fig.(2) and fig.(3), a comparative analysis of QCD transverse mo-mentum cutoff with respect to the energy is summarized in the followingtable. Hereby, based on the Bjorken scaling in QCD cutoff, we have clas-sified the nature of the transverse momentum, i.e., the energy emitted orabsorb in due course of the gluon density fluctuation.

Following the above observation as in fig.(2, 3), we provide the qualitativeanalysis of the energy emitted or absorb during the formation of parton inthe space of QCD transverse momentum cutoff and QCD renormalizationscale in fig.(4) with the energy definition as

lnEi,j =1

2c2

(log(

qiLj

)(1− exp(− 1

log( qiLj

))))2

, (22)

20

Figure 4: The emitted or absorb energy E plotted in the logarithmic scaleas a function of the transverse momentum q on Y -axis and L on X-axisdescribing the nature of the energy fluctuations by considering simultaneousvariations over q and L.

where qi ∈ (0, 10) and L ∈ (0, 1). Here, we find that most of the emissionshappen for a large QCD transverse momentum and small QCD renormaliza-tion scales. In other words, we find that there is almost no energy changefor a small transverse momentum and large QCD renormalization scale. Thecorresponding behavior as shown in fig.(4), where the energy augments to theorder of a fixed value in its logarithmic scale as the transverse momentumtakes a large value of the order 10 for a given small QCD renormalizationscale L. It is anticipated that the phase transition is expected to arise neara small value of the transverse momentum and QCD renormalization scale.Such processes are truly quantum in their nature that we leave it open for afuture research investigation.

In fig.(5) with x = 0.1, g = 10, k = 1, nf = 3, we provide a qualitativebehavior of the gluon density fluctuations in the space of the initial transversemomentum Q and QCD renormalization scale L. Here for B = 25, thediscriminant ∆ takes a large negative value of the order 8× 1010. We noticethat the value of the amplitude depends on the choice of the prefactor B.From the functional definition of the fluctuation discriminant

∆(i, j) = − 16B2

q(i)2l(j)2

(log(q(i)2l(j)2

))−4, (23)

21

Figure 5: The discriminant A := ∆ plotted as a function of the transversemomentum q on Y -axis and L on X-axis describing the nature of the energyfluctuations by considering simultaneous variations over q and L.

we observe that the sample becomes ill-defined when the initial transversemomentum, the minimum transverse momentum Q and QCD renormaliza-tion scale L take an identical value. In this case, from fig.(5), we see that thefluctuation theory analysis needs to be prolonged with a higher scale to checkthe stability of the configurations under fluctuations of the gluon density inthe space of {Q,L} . Such studies we leave open for a future research.

6 Conclusion and future directions

In this paper, we examine an extrapolated equivalent description of gluondensity fluctuations with respect to the initial transverse momentum andQCD renormalization scale fluctuate in the region where the proton has aneffective radius in the transverse momentum space. This offers an intrinsicstatistical understanding of overall structure of soft interaction and diffrac-tion dissociation phenomena. We also addressed some of the shortcomings ofthe SIBYLL 2.1 by considering an ensamble of QCD improve parton modelsand analysing beyond the nucleus-nucleus collisions using semisuperpositionmodel and the full Glauber model. Our analysis can as well be applied to un-derstand the antibaryon production,namely, the distribution of anti protonsin the central region of collisions Alexopoulos et al.(1996).

In this concern, we anticipate exploring the dependance of impact param-eter of the collision governing the relevant gluon density of the target nucleus

22

Alexopoulos et al.(1996). Such an understanding following our statisticalanalysis could improve the modelling of minijet model as a constraint mini-jet model with a refined profile function, whose analysis is relegated for a laterpublication. Note that our analysis entails the properties of hadron-nucleusand nucleus-nucleus interactions as achieved in Glauber setting Glauber etal.(1970). using two channel model, whereby we offer statistical propertiesof proton-proton interactions.

It would be interesting to examine the fluctuation theory based under-standing of rapidity and Feynmann distribution as in NA49 experiments Gor-naya et al.(2018).and pseudorapidity and multiplicity distribution of chargedparticles in central region. Indeed, our model by no means gives the final an-swer and most likely exhibits short comings in the limits where the statisticalapproximation breaks down.

With the above advantages and disadvantages of the double leading log-arithmic approximation of the gluon density function,we emphasize that ourstatistical analysis is able to successfully reproduce the increasing nature oftransverse momentum as energy scale is varied in many of experimental re-sults. This includes that energy scales of diffraction dissociation particlessuch as Pomerons, soft interaction models in the light of Regge theory, mini-jet productions,string fragmentation of quark-quark and quark-diquark pairs,CDF, P238, UAS, GEUSS,and UA4 collaborations (Nik et al.2018, Zha etal.2018, Szabolcs et al.2018, McNelis et al.2018, Khoze et al.2018). We fur-ther believe that application of our fluctuation analysis theory would offer animproved understanding of other models in analysing the behaviour of thevery high energy particles such as the analysis of extensive air shower andcosmic rays.

The study of multiple interacting soft systems composed of the hadron-hadron collisions with given profile functions play an important role. Namely,the proton-proton collision profile function is anticipated by using fuzzyGaussian soft interactions with an exponential form factor correspondingto the Gaussian profile function of the constituent protons, see Bruyn etal.(2018). for monojet and trackless jets as the final states towards develop-ments of the CMS Detector at the LHC. Such a model can be obtained asan effective configuration of the two body interaction system. Therefore, ex-ploring the stability of gluonic matter in the soft/ hard limit as a function ofthe energy and associated physical meaning in the limit of vanishing impactparameter is in the future scope of this research. Namely, for a given data,we wish to experimentally characterize an optimal value for the transversesize of a proton at an energy scale

√s for fixed values of initial energy s0

and Regge slope α′. In nutshell, because of multi-particle interactions, weanticipate that an energy dependent transverse momentum as function of

23

the energy square s and parameters of the chosen fluctuation dynamics as arandom observation sample. Such an analysis lies in the future scope of thisresearch.

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