A statistical model to calculate inclusive hadronic cross sections
aa r X i v : . [ nu c l - t h ] F e b EPJ manuscript No. (will be inserted by the editor)
A statistical model to calculate inclusive hadronic cross sections
G´abor Balassa and Gy¨orgy Wolf Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, H-1525 Budapest, HungaryReceived: date / Revised version: date
Abstract.
Hadronic cross sections are important ingredients in many of the ongoing research methods inhigh energy nuclear physics, and it is always important to measure and/or calculate the probabilities ofdifferent types of reactions. In heavy-ion transport simulations at a few GeV energies, these hadronic crosssections are essential and so far mostly the exclusive processes are used, however, if one interested in totalproduction rates the inclusive cross sections are also necessary to know. In this paper, we introduce astatistical-based method, which is able to give good estimates to exclusive and inclusive cross sections aswell in the energy range of a few GeV. The method and its estimates for not well-known cross sections, willbe used in a Boltzmann-Uehling-Uhlenbeck (BUU) type off-shell transport code to explain charmoniumand bottomonium mass shifts in heavy-ion collisions.
PACS.
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Hadronic cross sections at a few GeV are usually used asinput parameters in heavy-ion transport codes, where upto 10 −
20 GeV the main degrees of freedom are the knownbaryons, mesons and the well-established resonances [1,2,3,4]. In these simulations mostly 2 → N → M reactions as well [5,6]. It is not straight-forward how to include these types of collisions into thecodes, so these are mostly omitted in the simulations. Athigher energies, partonic degrees of freedoms are also be-coming important [7]. Recent calculations show that thecharmonium states e.g. J/Ψ, Ψ (3686) , Ψ (3770) will acquiremass shifts during their evolution in dense matter, whichcould indicate a non-zero value for the gluon condensateat specific nuclear densities [8,9,10]. This information, ifmeasured could be really important to non-perturbativequantum chromodynamics. For this purpose the inclusivecharmonium production cross sections should be known,however, they are not well (or not at all) measured inthe energy range we are interested in (2-10 GeV), so ithas to be estimated from some theory. The method pre-sented in this paper is able to predict inclusive cross sec-tions, which will be ultimately used to estimate charmo-nium production cross sections. The paper is organizedas follows. In sections one and two a basic formulationof the method and its capabilities to estimate exclusivecross sections are described. This can be found more rig-orously in [11], where the main idea and some exampleswere also given. After the basic formulation, section threededicated to the normalization and model error estima-tion. The fourth section describes a method, which is usedto calculate inelastic cross sections without summing over all the possible 2 → N reactions and using only 1-, or2 fireball event ratios. The model error is also addressedin this section. The fifth section divided into two subsec-tions. In the first subsection six examples are given for theinclusive cross sections compared to their measured val-ues, while the second subsection corresponds to charmo-nium production and the estimation of the charm quarkcreational probability. Finally the last section briefly sum-marizes the paper. The method is based on the so-called Statistical Bootstrap[12,13,14] approach, which idea was widely used to esti-mate particle multiplicities at high energies [15,16,17]. Inour model, we assume that at the initial state of a collisiona strongly interacting, compound system is formed (a so-called fireball), which will ultimately decay into hadronsgiving some specific final state of the collision. In the re-action probability, the initial fireball formation stage andthe hadronization stage can be separated as it is shownin Eq.1, where we assume that the first stage is describedby the total or inelastic cross sections of the reaction andthe second stage is described by some mixed statisticaland dynamical factors. σ n → k ( E ) = Z n Y i =1 d p i R ( E, p , ..., p n ) ! × Z k Y i =1 d q i w ( E, q , ..., q k ) ! (1) In our model it is the inelastic cross section is used, ratherthan the total due to the exclusion of the elastic channel. G´abor Balassa, Gy¨orgy Wolf: A statistical model to calculate inclusive hadronic cross sections where E is the CM energy of the collision ( E = √ s ), σ n → k is the generalized n-body cross section, p , ...p n arethe momenta of the incoming particles, q , ...q k are themomenta of the outgoing particles, R ( E, ... ) is the func-tion describing the initial state dynamics, and w ( E, ... )describes the probability of a specific k-body final state.At the first stage of the hadronization process the fireballwith invariant mass M = √ s directly giving hadrons ordecay into two subsequent fireballs with smaller masses m , m where m + m = M should be fulfilled. The re-sulting fireball will hadronize or decay into further fire-balls. At the end of the chain, all of the fireballs have tohadronize leaving only hadrons at the end. The proba-bility of the one-, two, three-, etc... fireball scheme canbe calculated with an appropriate model described in ourprevious paper on this topic [11]. The possible number ofhadrons coming from one fireball is two or three with theprobabilities of P d = 0 . P d = 0 .
24 which is describedin the original formulation of the statistical bootstrap byFrautschi [18].An extra ingredient to the full probability (not men-tioned in the original formulation, because the normal-ization was cancelled in the calculated ratios), which isneeded to make a proper normalization is to separate sta-ble and non-stable/resonant particles and use a weightingfactor to each of the channels corresponding to the fullwidth of the particles. In the model, the stable particlesare the ones, which could not decay into other hadrons viathe strong interaction. Examples are the N ∗ nucleon reso-nances, which also could decay strongly to protons/neutrons,so they are not stable. These unstable particles are weightedby a relativistic Breit-Wigner factor[19]: F BRi ( E i , m i ) = 1 π E i Γ i ( E i − m i ) + E i Γ i (2)where m i is the i ’th particle mass, Γ i is its total width, E i is the invariant mass of the resonance, and the energyintegral of F BRi ( E i , m i ) is normalized to unity in E ∈ [ −∞ , ∞ ].The appearance of the Breit-Wigner factor can be ex-plained by the propagator of a particle with finite but nonnegligible width, when calculating Feynman diagrams e.g.as it was described in [20]. To calculate it’s contribution tothe full probbility it has to be integrated out in the phys-ical region, defined by the masses of the possible decayproducts of each resonance.Putting together everything the full probability of aone-fireball scheme, which hadronizes into two or threeparticles can be calculated as it is shown in Eq.3: W n i ,i ( E ) = P fb ( E ) C Q i ( E ) P H,in i ( E ) P j C Q j ( E ) P H,jn j ( E ) == P fb ( E ) T i ( E ) P j T j ( E ) , (3) There is also a possibility that one fireball could decay intomore than three particles, however due to the small probabili-ties, they are neglected in our model. where P fb is the probability of the one fireball decayscheme, C Q i is the quark combinatorial probability forfinal state i , which is proportional to the number of quark-antiquark pairs created at energy E , and to the u, d, s, c, ... quark creational probabilities fitted from measured data.Furthermore, n i stands for the number of hadrons comingfrom the i ’th fireball (2 or 3) and P H, n i is the hadroniza-tion probability factor given by Eq.4 and Eq.5 for two-and three-body states respectively: P H,i ( E ) = Y l =1 (2 s l + 1) P d Φ ( E, m , m ) ρ ( E )(2 π ) N I ! (4) P H,i ( E ) = Y l =1 (2 s l + 1) P d Φ ( E, m , m , m ) ρ ( E )(2 π ) N I ! , (5)where P d = 0 .
69 and P d = 0 .
24 are the probabilities ofa two or three particle final state decays, s l is the spinof the l ’th particle, N I is the number of same particles inthe final state, m i are the masses of particles, Φ n ( E ) isthe n -body phase space integral, ans ρ ( E ) is the densityof states (DOS) given by the statistical bootstrap [21,22]: ρ ( E ) = a √ E ( E + E ) . e ET , (6)where a , E and T are free parameters, however a alwaysfactorized out due to the normalization and E = 500MeV is previously fitted to the experimentally measuredDOS [23]. In our previous paper we also set T = 160 MeVusing calculated and measured cross section ratios.All of the factors in Eq.3 are described in detail in [11].To simplify the following sections we have introduced thenotation: T i ( E ) = P fbi ( E ) C Q i ( E ) P H,in i ( E ) (7)The two- and three-body phase-spaces are weighted bythe product of the particles Breit-Wigner factors (if it is aresonance) and integrated out to the corresponding energyregion as it is seen in Eq.8: Φ k ( M, m , .., m k ) = V k − × (cid:16) Z Y r ∈ R dE r (cid:17)(cid:16) Z k Y i =1 d q i (cid:17) Y r ∈ R F BRr ( E r , m r ) × δ (cid:16) k X j =1 E j − M (cid:17) δ (cid:16) k X j =1 q j (cid:17) (8)where q j is the three momenta of the j’th particle, E i is theenergy of the i’th particle and F BRr ( E r , m r ) is the Breit-Wigner probability factor of the resonance r , and V is theinteraction volume. If all of the particles are stable thenthe integrals in the first bracket and the Breit-Wigner fac-tors are not needed. Otherwise the r index goes through allof the resonances. The Breit-Wigner factor will decreasethe phase-space according to the resonance width e.g. fora three-body channel with one resonance and two stable ´abor Balassa, Gy¨orgy Wolf: A statistical model to calculate inclusive hadronic cross sections 3 E [GeV] ( E , m S , m R , m R ) / ( E , m S , m S , m S ) Fig. 1.
Ratio of the three-body phase-space integrals for oneresonant and two stable vs. three stable particle configurationswith m = m = m = 1 GeV masses, and Γ = 0 . S index in m iS stands for ”Stable”,while the R index stands for ’Resonant’ particles. particles ( m = m = m = 1 GeV and Γ = 0 . p ¯ p and π − p reactions due to their different quantumnumbers. The extension to more fireballs is not straight-forward, due to the different possibilities of how to includequantum number conservation into the model. In our nor-malization scheme, only the full final state has to respectthe conservation laws and the quantum numbers of thedifferent fireballs could be anything, there are no restric-tions to it. The normalization procedure for more thanone fireball is straightforward to see from the two fireballprobability in Eq.9: W n i ,n j ,ij ( E ) = P fb ( E ) N i,j ! R x max x min dx T i ( x ) P a T a ( x ) T j ( E − x ) P a T a ( E − x ) Z ( E ) (9)where the normalization sum Z ( E ) is described by Eq.10: Z ( E ) = X
0) (baryon num-ber, charge, strangeness, charmness). For the two fireballnormalization sum, we only consider the channels whichat the end give back the initial quantum numbers e.g. Q = (1 , , − ,
0) and Q = ( − , , ,
0) is a good choiceas S = Q + Q . So separately the fireballs do not haveto respect the conservation laws, but the whole systemhas to and give back the initial quantum numbers. It isimportant to note that the N-fireball probability can bebuilt up by the 1-fireball factors and their correspondingintegrals as it can be seen in Eq.11 for the general casewith N-fireballs. W k,i ..i k ( E ) = P fbk ( E ) 1 Z k N i ,..i k ! Z x max x min k Y a =1 " dx a ×× T i a ( x ) P j T j ( x ) δ (cid:16) k X a =1 x a − E (cid:17) (11)where Z k ( E ) is the energy-dependent k-fireball normal-ization factor given by Eq.12. Z k ( E ) = X
15. These fits were made without includingcharmed particles, however due to the smallness of thecharm creational probabilities, these values are approxi-mately still valid in the energy range, we are interestedin. For the heavier charm quarks and the correspondingcharmonium particles, the fit P c is not straightforward asthe experimental cross sections are very limited and thecross sections are too small to use the same method whichis used to fit P u , P d , P s . The solution to this problem isdescribed in Sec.5.2.There are some limitations to the model, however, whichhas to be addressed. It is not suited to describe the elasticpart of the two-body reactions, so that channel is simplyomitted from the calculations and the inelastic ratio isused instead of the total cross section. Another limitationis that the model cannot describe the A + B → C reac-tions, where A, B, C are some particles. This is the con-sequence of the Frautschi picture of the Bootstrap, wherethe minimum number of final state hadrons are 2. Thisproblem is solved by introducing an extra contribution tothe cross section described by a Breit-Wigner cross sectionformula σ BR ( E ): σ BR ( E ) = 2 s C + 1(2 s A + 1)(2 s B + 1) 4 πp i E Γ C → AB Γ tot ( E − m C ) + E Γ tot (13)where A, B are the colliding particles, C is the createdresonance, which could decay into some specific final statedescribed by it’s total decay width Γ tot . The s A,B,C fac-tors are the spins of the corresponding particles, and p i is the center of mass momentum of the initial state. Therelevant decay widths can be gathered from the ParticleData Book (PDG) [24]. With this in mind the probabilityof a specific channel corresponding to the inelastic crosssection is expressed in Eq.14. σ i ( E ) σ inelastic ( E ) = R i ( E ) + P j σ BRj ( E ) σ inelastic ( E ) (14)where R i ( E ) is the cross section ratio from our statisticalmodel, where all the possible fireball decay probabilitiesare summed over (Eq.15). R i ( E ) = N X k =1 W n i ..n ik k,i ,..i k ( E ) (15)where the sum of all the possible k = 1 ..N fireball schemesfor a specific final state are taken. An example is the fi-nal state 3 π + π − where one has to sum the 1-,2-, and 3 fireball contributions as well. If one wants to calculate in-elastic cross sections the full normalization sum has to becalculated and possibly an uncertainty analysis has to bedone, which is the main focus of the next section. The normalized probability in Eq.4 should give back the σ I /σ inelastic cross section ratio by definition. If we havea specific channel ” i ” which only could come from a onefireball decay scheme, then only the one fireball normal-ization sum should be calculated. The following four pro-cesses were calculated to test the method: – p ¯ p → π + π − – p ¯ p → n ¯ n – p ¯ p → ∆ ++ ∆ −− – p ¯ p → K + K − As the measured cross sections and the model also couldcontain errors it is possible to estimate an error distri-bution of the model from the distribution of the relativeerrors. To this purpose, the relative error is calculated asit is shown in Eq.16. k = r − W r , (16)where r = σ i /σ inel and W is the one fireball model prob-ability. The model calculations does not contain A + B → C → D + E reactions, due to the lack of knowledge ofthe branching ratios of heavy mesons to proton and an-tiproton. Using the Gaussian error propagation formula,considering that ∆σ i = 0, ∆σ inel = 0, and ∆W = 0the absolute error of the measured relative error ∆k isexpressed in Eq.17. ∆k = W r (cid:16) ∆r r + ∆W W (cid:17) (17)The relative error of the model could be expressed fromEq.17 as follows (Eq.18): ∆W W = r ∆kW − ∆r r (18)If the ∆k uncertainty is approximated by the standarddeviation of the relative error distribution, and assumingan energy-dependent model error, the relative model er-ror distribution could be expressed from Eq.18 and itshistogram can be seen in Fig.2:The estimated relative error of the model is calculatedfrom this distribution by taking its mean value, which isapproximately ∆W /W ≈ .
4. This value will be theapproximated relative error of the model, which will beused in the following sections, where the inelastic crosssections are calculated. The calculated model probabilitywith the measured ratios can be seen in Fig.3, where theuncertainty bound is also calculated from the previouslyestimated model error. It is again worth to mention, that ´abor Balassa, Gy¨orgy Wolf: A statistical model to calculate inclusive hadronic cross sections 5
Fig. 2.
Estimated relative error distribution of the model.
E [GeV] + - / i ne l -3 measurementmodeluncertainty bound Fig. 3.
The process p ¯ p → π + π − ratio to the inclusive crosssection σ inelp ¯ p with the full normalization for one fireball. Datataken from [25][26]. in this calculations the possible pp → X → π + π − pro-cesses are neglected, where X is a heavy meson abovethe two proton threshold. This is mainly due to the lackof knowledge of the branching fractions at the PDG forheavy meson decays to proton and antiproton. In heavy-ion simulations, if the interesting quantities areparticle multiplicities, then the desirable cross sections arethe inclusive ones e.g. if one interested in the charmoniumspectra which is measured by their dilepton pair decays aninteresting background process could be X → D ¯ D , whereeach D meson could decay into an electron(positron) anda Kaon, giving an overall dilepton pair at the end. In thisexample, the ( X → D ¯ D + anything) process should be cal-culated as every possible channel is giving an extra contri- bution to the full background. The naive way to calculateinclusive cross sections to sum over all the possible reac-tions, which respects the quantum number conservationlaws, however, this method is not very efficient and thereis a much easier way to do this.Let us take the probability ratio of a channel with twodifferent normalization, which only means the normaliza-tion sum contains different channels. The inelastic sumcontains all the possible final states allowed by quantumnumber conservation, while the inclusive sum contains ev-ery possibility, but with one specific particle fixed in ev-ery possible final state. To make things easy the referencechannel should be strictly 1 , , , ...N fireball channel with-out mixing so that the normalization is clear. Let us takea reference channel I , which is coming from strictly onefireball channel and calculate the ratio like in Eq.19: σ I /σ inelastic σ I /σ inclusive = σ inclusive σ inelastic = P j ∈ inclusive T j ( E ) P j ∈ inelastic T j ( E ) (19)where in the ratio the common factors e.g. P fb are droppedout and only the normalization terms in the denominatorremained. The Breit-Wigner factors are also considered inthe phase-space integrals. In this way the inclusive crosssection can be expressed as it is shown in Eq.20: σ inclusive = σ inelastic P j ∈ inclusive T j ( E ) P j ∈ inelastic T j ( E ) (20)A problem with using the one-fireball ratio is the small-ness of P T i at higher energies, which means that theprobabilities at higher energies for all of the one fireballpossibilities are negligible or practically zero. A solutionto this problem is to use the two-fireball ratio due to theirnon-negligible normalization sums even at the energies upto 15 GeV. The inclusive cross section from strictly two-fireball channels can be expressed in the same way withthe corresponding normalization sums as in Eq.21: σ inclusive = σ inelastic × (21) × " X ij ∈ S inclusive N ij ! Z dxH i ( x ) H j ( E − x ) // " X kl ∈ S inelastic N kl ! Z dxH k ( x ) H l ( E − x ) where P ij ∈ S inclusive means every possible two-fireball finalstate from the inclusive set, which respects the conserva-tion laws of the initial state, and for simplicity, we defined H i ( x ) as: H i ( x ) = T i ( x ) P j T j ( x ) (22)Typically inclusive cross sections like p ¯ p → I + X areneeded where I is a fixed particle. The inclusive sum thenincludes all of the possible 2 fireball final states, whichcontains at least once particle I in its final state, how-ever, it is also possible to calculate inclusive reactions like p ¯ p → I + J + X , where I and J are two fixed particles and G´abor Balassa, Gy¨orgy Wolf: A statistical model to calculate inclusive hadronic cross sections X is everything else. The generalization is straightforwardand the only task is to separate the possible 2 fireball re-actions where we have a specific number of fixed particles.For one fireball scheme, the calculation is simple and it ispossible to include every possible two and three-body finalstates into the sum with little effort, however, if one wantsto calculate the ratio from more than one fireball decaysthe possible number of channels are too huge and the inte-gration would take many hours on a standard laptop. Dueto the numerical complexity, except for one case, only theone fireball calculations are shown in this paper, however,it can be shown that if one wants to lower the uncertaintyof the model, then taking the two fireball normalizationratio is a good option to do so. To see this let us assumethat each H i ( x ) is a uniformly distributed random vari-able with some mean µ > . µ .Taking the two-fireball normalization ratio the followingdistribution has to be calculated (Eq.23): f ( E ) = Z dx M X i,j =1 g i ( x ) g j ( E − x ) Z dx N X i,j =1 g i ( x ) g j ( E − x ) , (23)where g is a uniformly distributed random variable, and M < N because M corresponds to the inclusive sum,and N corresponds to the inelastic sum. For simplicitylet us assume that each g i is energy independent so that f ( E ) = f will also be energy independent. To furthersimplify the problem the ratio in Eq.24 is calculated: f = P Mi =1 g i P Ni =1 g i , (24)where g i is now constant, so the integral can be factoredout, and instead of taking every possible i, j combinations,we simply take the square of the generated g i randomvariables. This will not change the main conclusion, so forthe first estimation, it is sufficient to see the relative errordistribution of the calculated ratio in Eq.24. The relativeerror distribution is calculated in the following steps:1. Generate N number of uniformly distributed U [0 , → g i samples. These will be the noiseless data.2. Add a random noise to each of the g i ’s so that g i = g i + U [ g i − . g i , g i + 0 . g i ] will be the data with arelative error of 0.5.3. Take M number of samples from the previously gen-erated samples. These will correspond to the inclusivesum.4. Calculate the ratios with the noiseless samples f , andwith the noisy ones f .5. Calculate the relative error r = | f − f | /f Do the previous steps many times so at the end a rela-tive error distribution from the calculated r ’s is obtained.As has been done previously, the estimated relative errorwill be the mean of the obtained relative error distribu-tion. The mean relative error dependence on the number Fig. 4.
The relative error dependence of the two fireball nor-malization ratios on the number of inelastic-, and inclusivechannels (N,M). of inelastic channels (N) i = 1 .. j = i..
100 can be seen in Fig.4.The number of inelastic channels has to be larger thanthe number of inclusive ones e.g.
N > M . The relativeerror at low multiplicity is around 0.5, which means theaccumulated relative error is the same as the individualrelative errors. If the multiplicity in the numerator and inthe denominator increasing the accumulated relative errorwill be smaller. The main conclusion from this simple sim-ulation is that if one wants to make the estimated relativeerror smaller it is sufficient to calculate e.g. the two fireballnormalization sums instead of the one fireball sums, how-ever, its numerical complexity is much greater. Also, thetwo fireball cases will cover a different energy range due tothe different thresholds and different processes considered.
To validate the theory six channels were calculated withthe formalism showed in Sect. 3. In three process the onefireball probability ratios were calculated and comparedto the measured cross section ratios, while in one case thetwo fireball ratio is calculated. The errors of the measured-and the calculated cross section ratios are also shown. Thefirst reaction is the pπ − → ρ X , where X is any 1 , , ...N number of particles, which allowed by the quantum num-ber conservation of the initial pπ − collision, and is allowedby the specific number of fireballs we used in the calcu-lations. To calculate the ratio, first, the inelastic sum hasto be done. The second step is to find all the resonanceswhich could decay into ρ e.g. N → ρN and selectonly those channels into the sum (with the direct ρ X channels as well). These are most of the ∆ -s and nucleonresonances and are taken from the PDG, where only par- ´abor Balassa, Gy¨orgy Wolf: A statistical model to calculate inclusive hadronic cross sections 7 ticles that have at least 3 stars are included. The resultscan be seen in Fig.5.In the second example, the process pπ − → K X is cal-culated with one- and also with two fireball ratios in orderto compare the results. In the inclusive sum, the particleswhich could decay into K -s have to be included. Theseare some of the N, Λ , Σ , and K ∗ resonances. The inelas-tic sum has to include all processes which have one baryonnumber, zero strangeness, zero charmness, and zero charge.The ratios from the two- and one fireball processes areshown in Fig.6, where in both cases a really good agree-ment with the measured data is achieved. The error inthe two fireball case is calculated from the simulation de-scribed in Sec.4, using the number of inelastic, and thenumber of inclusive channels.The third example is the channel p ¯ p → ρ X , where theinelastic sum will be different than in the previous tworeactions because in p ¯ p collisions all the important con-served quantum numbers are zero. The results are shownin Fig.7, where again a really good agreement is achievedeven at higher energies (15 GeV).The fourth calculated process is the pp → ρ X inclu-sive channel, where the inelastic sum includes all the pos-sible final states with quantum numbers (baryon number,charge, strangeness, charmeness)=(2,2,0,0). The inclusivesum contains all the processes where at least one ρ mesonis present and also all the relevant resonances which coulddecay to ρ . The results are shown in Fig.8.The final non-charmed examples are the strange vectormeson K ∗ (892) + and K ∗ (892) − production in π − p colli-sions. The strange vector meson production is an impor-tant tool to study the dense hadronic matter in heavy ioncollisions [27][28][29] due to their spectral function depen-dence on the nuclear density and temperature. We expectthe model to give back the low energy suppression of the K ∗ (892) − state to the K ∗ (892) + meson. The results canbe seen in Fig.9, where the upper side shows the K ∗ (892) + ratio to the π − p inelastic cross section and the lower sideshows the K ∗ (892) − one. Also some measured values areshown from [25][26]. It can be deduced that the main fea-ture ( K ∗ (892) − suppression) can be described quite wellwith the model. It should be worth noted however thatfor the strange quarks, we used a constant value for the P s quark creational probability, fitted from p ¯ p → K + K − process at the energy range E ≈ [1 . , .
6] GeV, which isa simplification, as the strange quark suppression shouldbe energy dependent, so does P s . Nevertheless, even withthis assumption the model gave back the main qualitiesof the cross sections.In every aforementioned process, a really good matchis achieved with only one fireball process considered. Usingtwo fireball ratios the match is even better, which corre-sponds to the smaller model error estimated in the previ-ous section. One of the models aim is to give estimates to charmo-nium creation cross sections below 10 GeV, which will be
E [GeV] i n c l u s i v e / i ne l a s t i c MeasurementModelUncertainty bound
Fig. 5.
The process pπ − → ρ X ratio to the inclusive crosssection σ inelpπ − . Data taken from [25][26]. E [GeV] i n c l u s i v e / i ne l a s t i c MeasurementModel (2 fireball)Uncertainty bound
E [GeV] i n c l u s i v e / i ne l a s t i c MeasurementModel (1 fireball)Uncertainty bound
Fig. 6.
The process pπ − → K X ratio to the inclusive crosssection σ inelpπ − from two- and from one fireball processes. Notethe smaller uncertainty bound in the two fireball case. Datataken from [25][26]. very useful inputs for upcoming transport simulations, re-garding charmonium mass shifts in heavy ion collisions.To make reliable estimates the model should describe wellthe available low energy cross sections, which are very rareat the moment. A good collection of data are used for fit-ting in [30]. In our model, the necessary parameter, whichhas to be fitted from experiments is the charm quark cre-ational probability ( P c ), which is used to calculate thequark combinatorial factors. Due to the much larger massof the charm quark than the u,d,s quarks, this value are ex-pected to be much smaller then the others, thus to createmore than one charm-anticharm pair is negligible in thecalculations. Furthermore we expect it to increase withenergy, rather than to be a constant value. In contrastwith our previous attempt in [11], where we tried to fix aconstant P c value at one point using a fit from [30] near G´abor Balassa, Gy¨orgy Wolf: A statistical model to calculate inclusive hadronic cross sections
E [GeV] i n c l u s i v e / i ne l a s t i c MeasurementModelUncertainty bound
Fig. 7.
The process p ¯ p → ρ X ratio to the inclusive crosssection σ inelp ¯ p . Data taken from [25][26]. E [GeV] i n c l u s i v e / i ne l a s t i c MeasurementModelUncertainty bound
Fig. 8.
The process pp → ρ X ratio to the inclusive crosssection σ inelpp . Data taken from [25][26]. threshold, with the calculation of one exclusive channel,now we have the method to fit for measured inclusive dataat higher energies. This was one of the main reasons weextended the model to describe inclusive cross sections.For the quark creation probability, we expect a linearrise with energy with the simplest assumption possible P c = aE , where only one parameter, the slope a has to bedetermined. This parameter enters into the quark num-ber probability mass function described by a multinomialdistribution shown in Eq.25 F ( N, n i ) = N ( E )! Q i = u,d,s,c n i ! Y i = u,d,s,c P n i i , (25)where N ( E ) = 1 + p E /T , (26) E [GeV] -2 i n c l u s i v e / i ne l a s t i c Measurement (K* - )ModelUncertainty bound E [GeV] -2 -1 i n c l u s i v e / i ne l a s t i c Measurement (K* + )ModelUncertainty bound Fig. 9.
The processes pπ − → K ∗ (892) + X and pπ − → K ∗ (892) − X ratio to the inclusive cross section σ inelpπ − . The up-per half of the figure shows the K ∗ (892) + production, whilethe the lower half shows the K ∗ (892) − case. Data taken from[25][26]. is the total number of quark-antiquark pairs [31], withhadronization temperature T , and P i is the quark cre-ational probability for the i = ( u, d, s, c ) type quarks.The expected number of quark-antiquark pairs of type i , will be n i = P i N , which corresponds to the maximumof the probability mass function. Note that the bottomquark is still missing, and we intend to fit the correspond-ing P b value as well in the future. The expected num-ber of different quarks and antiquarks then build up thehadrons in all possible ways described by simple combina-torics, multiplied by the corresponding probability fromthe probability mass function, and at the end are normal-ized with all the possible final states, giving a final prob-ability to create one specific hadronic final state. As thecharm quark creation is expected to be highly suppressedonly n c = 1 is interesting at the moment, which will cor-responds to a constrained maximum value of F ( N, n i ) incontrast to the non-constrained maximum, where n c = 0.The suppression ratio is then proportional to the charm-anticharm creational probability, so it can be easily fit-ted using the suppression to the non-charmed channels.To see this, let us assume that the one charm-anticharmquark pair is created in expense to one strange-antistrangequark pair, so n ′ c = 1, and n ′ s = n s −
1, while n ′ u = n u and n ′ d = n d . The suppression ratio calculated from theprobability mass function is then: γ c = F ( N, n ′ u , n ′ d , n ′ s , n ′ c ) F ( N, n u , n d , n s , n c ) = n s ! P c ( n s − P s = P c N (27)The derived formula for γ c expresses the fact, that evenwith a constant P c value, the suppression will be energydependent if γ c <
1, as the total number of quark-antiquarkpairs N are also energy dependent. After γ c reaches 1 thesuppression from the probability mass function is gone, sono need to constrain the distribution anymore. The ex- ´abor Balassa, Gy¨orgy Wolf: A statistical model to calculate inclusive hadronic cross sections 9 pected number of charm-anticharm quarks then will be n c = P c N , so the transition between the suppressed andnon-suppressed probability is continous. The vanishingsuppression ratio here only means, that there will be nomore suppression from the constrained probability massfunction, however if P c is still smaller than the other quarkcreational probabilities (which will be the case up to a few100 GeV), then the charm production is still suppressedcompared to the up, down and strange quarks.As we assumed a linear relationship between P c andthe energy, the γ c suppression ratio can be expressed as: γ c = a E + E p E /T , (28)where a is the slope parameter, which has to be deter-mined from experiments. Using experimental charmoniumproduction cross section data in pN collisions, this pa-rameter is determined to be a = 0 . − (see be-low), so the charm-anticharm quark creational probabilityis given by P c = 0 . · E . With the inclusion of anothertype of quark the previous P u , P d , P s fits were also haveto be modified. In the simplest case, we can assume that P ′ i = P i − P c / i = ( u, d, s ), so that P ′ u + P ′ d + P ′ s + P c = 1holds, however due to the small value of P c in the en-ergy range below a 100 GeV, we get P ′ u ≈ P u , P ′ d ≈ P d , P ′ s ≈ P s , so the previously used values for the u, d, s quarkscan still be used in this energy range.To calculate the cross sections the two fireball prob-abilities are used, as it gives smaller uncertainties, withthe price of taking much longer time to calculate the nor-malization factors. We intend to describe the available lowenergy pN and πN data with the model for which, we se-lected some experimental points below 10 GeV from thecollection in [30]. At this stage the uncertainties in theexperiments are not really important as the main goal isto describe the 2-3 times larger cross sections of the πN collisions in contrast to the pN collisions. To this end, andto check the model predictability, only one measurementpoint is used to the fit from the pN data at E ≈
10 GeV,which is enough, as we only have one free parameter in P c ,namely a . The tuning was done by hand to approximatelygive the correct value at the one point used for the fit. Forthe cross sections the decays from Ψ (3686), χ c and χ c mesons are also included, however as there are no bottomquarks in the current model the decays from hadrons con-taining b quarks are not included. After the tuning of theslope parameter, the full cross sections were calculated forthe pp and for the π − p case. The results can be seen inFig.10 and in Fig.11. The calculations show a good matchwith the data for both cases even with the simple form ofthe P c and with the restricted one point fit, thus it can beconcluded that the main feature of the πN enhancement isdescribed well in the model. To test the model capabilitiesfurther the proton-antiproton to proton-proton inclusive J/Ψ cross section ratio is calculated at E = 24 . E [GeV] -2 -1 pp J / X [ nb ] MeasurementModelUncertainty bound
Fig. 10.
Inclusive
J/Ψ production cross section estimates inproton-proton collisions. Only the rightmost measured pointnear 10 GeV is used to fit the slope in P c = aE . Data takenfrom [30]. E [GeV] -2 - p J / X [ nb ] MeasurementModelUncertainty bound
Fig. 11.
Inclusive
J/Ψ production cross section estimates in π − p collisions. Data taken from [30]. The method thus proved to be able to give satisfac-tory results to low/medium energy charmonium cross sec-tions, however putting more particles (e.g. B mesons) intothe model could change the higher energy parts of thecross sections, meaning the P c quark creational probabili-ties might have to be further fine tuned. Also for P c a verysimple linear assumption was made, and it could be inter-esting to give a theoretical background to its nature. Thisalso applies to the strange quark creational probability P s ,where we simply assumed a constant value throught thewhole energy range we are interested in. . ± . ± .
06 0 . ± . Table 1.
Inclusive
J/Ψ cross section ratio for ( pp → J/ΨX ) /(¯ pp → J/ΨX ) collisions at √ s = 24 . In this paper, a statistical model is introduced based onthe statistical Bootstrap approach, which is able to givereasonable good estimates to hadronic cross section ra-tios up to a few GeV energy range. The full normalizationsum is calculated for one and two fireball decay schemes.The cross section ratios for eight different processes werecompared to the model calculations. From the calculatedrelative errors, the model uncertainty is also estimatedgiving an overall energy independent relative error for theone fireball decay probability. The calculated normalizedprobabilities are in good agreement with the measuredvalues. The model is extended to give inclusive cross sec-tions from the ratios of the normalization sums, which isvalidated through six distinct inclusive non-charmed pro-cesses, namely, pπ − → ρ X , pπ − → K X , pπ − → K ∗ + X , pπ − → K ∗− X , pp → ρ X and p ¯ p → ρ X , where in eachcase a really good match with measured data is achieved.The charm quark creational probability is fitted with thehelp of the pN → J/Ψ X inclusive data, allowing us to giveestimates to charmed final states as well. The model isfurther validated through two charmed processes, namely pp → J/Ψ X and π − p → J/Ψ X , both giving good resultscompared to the experimental data. For a last applica-tion the ( pp → J/Ψ X ) / (¯ pp → J/Ψ X ) ratio is calcu-lated at √ s = 24 . References
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