Charmed and bottomed hadronic cross sections from a statistical model
aa r X i v : . [ h e p - ph ] F e b EPJ manuscript No. (will be inserted by the editor)
Charmed and bottomed hadronic cross sections from a statisticalmodel
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.
In this work, we extended our statistical model with charmed and bottomed hadrons, andfit the quark creational probabilities for the heavy quarks, using low energy inclusive charmonium andbottomonium data. With the finalized fit for all the relevant types of quarks (up, down, strange, charm,bottom) at the energy range from a few GeV up to a few tens of GeV’s, the model is now consideredcomplete. Some examples are also given for proton-proton, pion-proton, and proton-antiproton collisionswith charmonium, bottomonium, and open charm hadrons in the final state.
PACS.
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Low and medium energy (up to 20-30 GeV) inclusive andexclusive cross sections are important inputs for trans-port calculations in heavy ion collisions [1,2,3,4,5]. Mostof these cross sections are coming from measurements [6,7], and/or from some effective model calculations [8] e.g.Walecka type models [9]. For many interesting processes,measurements are nonexistent, or very sparse e.g. for strangevector mesons K ∗ (892) [10,11,12] or charmonium crosssections near threshold [13,14]. As some of these hardly ac-cessible processes could be important in the investigationof strongly interacting nuclear matter formed in heavy ioncollisions [15,16,17], phenomenological model calculationsor extrapolation from fits to available data are necessary.In [18] we proposed a model, based on the statistical Boot-strap approach [19,20,21], which were able to give reliableestimates to low/medium energy cross sections (few GeV)for exclusive hadronic processes. The model is extendedin [22] to be able to give estimatations to inclusive pro-cesses as well, with an approximated error bound comingfrom the model uncertainties. Using the extended model,we were able to fit the charm quark creational probabil-ity to existing charmonium data. Using the fitted value,the model had been proved to be able to describe in-clusive charmonium production in proton-proton, and inpion-proton collisions. In this paper, we further extend themodel by introducing more particles into the calculations,and by fitting the bottom quark creational probability.These extensions will slightly change the previously fittedvalue for the charm quark creational probability as well.Including heavy quarks into statistical models have beendone in the past [23], where usually a suppression factor isintroduced into the quarks momentum distribution func-tion, which measures the deviation from chemical equilib-rium. In our model the quark creational probabilities are taking over this role, which were fitted to measured crosssections, with final states containing heavy quarks. Withthe modifications the model is now considered complete,as the energy range, we intend to use is far below the topquark mass, so its inclusion in the model is not necessary.The paper is distributed as follows. In Sec.2 a basic formu-lation is given, which covers the main ingredients of themodel. In Sec.3 the fitting procedure for the charm andbottom quark creational probabilities is covered, while inSec.4 some charmonium, bottomonium and open charminclusive cross sections are calculated for proton-proton,proton-antiproton and pion-proton collisions. After theseexamples Sec.5 concludes the paper. The model is based on the assumption that during a col-lision process a so called fireball is formed, which after ashort time, will hadronize into a specific final state. Thecollision process therefore can be factorized into an initialdynamical part forming the fireball, and a mixed dynami-cal and statistical part, which describes the hadronization,as it is shown in Eq.1 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)where σ n → k is a generalized transitional probability foran n → k process, p i is the 3-momenta of the incomingparticles, E is the center of momentum energy of the n -body collision, and q i is the 3-momenta of the outgoing G´abor Balassa, Gy¨orgy Wolf: Charmed and bottomed hadronic cross sections from a statistical model particles. The function R ( E, p , ...p n ) describes the initialstage of the collision, while w ( E, q , ..., q k ) describes thehadronization of the fireball. Throughout this paper, weonly consider two-body collisions, so σ → k becomes thecross section of the collision. It is assumed that the dy-namical part is described by the inclusive cross section ofthe colliding particles, so the integral in the first bracketsimply becomes the inelastic cross section of the two-bodyreaction. This assumption is widely used in many statis-tical models in the past [24,25]. Although it is worth tomention that it is possible to include more than two-bodyreactions as well, where only the initial dynamical part R ( E, p , ..., p n ) has to be changed accordingly to sometheory. The integral in the second bracket describes thefireball hadronization, and will be denoted by W ( E ). Thehadronization probability depends on the following fac-tors: (1) Fireball decay scheme probability (2) Fireballdensity of states (3) Phase space of the final state (4)Quark combinatorial factor (5) Quark creational proba-bilities. In the followings a short description is given toeach of the ingredients.After the fireball is formed, there is a possibility thatthe fireball with invariant mass M hadronizes into somefinal state or decays into smaller fireballs with invariantmasses M , M . Each of these smaller fireballs are thenagain choose to hadronize or decay into smaller fireballs.At the end of the chain there will be a specific number offireballs, with invariant masses M , M , . . . M k , and eachof them ultimately decay into a specific hadronic finalstate. The probability of the number of fireballs is givenby kinematical considerations and is described in detail in[18]. The hadronization procedure is based on the statisti-cal bootstrap approach, which sets the probability for thenumber of hadrons coming from one fireball to P fb = 0 . P fb = 0 .
24 [26]. More particles have negligible proba-bilities compared to the two-, and three body cases, there-fore in our model the possible number of hadrons comingfrom one fireball is two, or three. However it is straightfor-ward that due to the many number of smaller fireballs thenumber of hadrons in the full final state could be muchmore than three.Following the bootstrap approach the hadronizationinto final state particles should depend on the density ofstates (DOS), and also the two-, and three body phasespaces with a slight difference on how we handle non-stableand stable particles. The density of states is given by: ρ ( E ) = a √ E ( E + E ) . e ET , (2)where T is the hadronization temperature, E = 500 MeVis a cut off parameter, and a is an irrelevant normaliza-tion factor, which disappears in the final results. Theseparameters are fitted using experimentally measured par-ticle multiplicities and momentum spectra for protons, pi-ons and kaons in [27], and in [19]. It is also mentionedthat T could be essentially anything from 130 MeV to170 MeV (with some corresponding changes in E and a ),giving almost the same mass spectrum, therefore, as ourmodel is not really sensitive to E and a , we did our own fit to T , using exclusive cross section ratios in [18], giving T = 160 MeV.In the model, stable and non-stable hadrons are dis-tinguished, which is manifested in the phase spaces con-sidered. A hadron is considered a resonance, if it has anon-negligible width and is able to decay into a lower ly-ing hadronic state via the strong interaction. Taking intoconsideration the two types of particles, the phase spaceis written as: 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 × δ (cid:16) k X j =1 E j − M (cid:17) δ (cid:16) k X j =1 q j (cid:17) (3)where q j is the three momenta of the j’th particle, E is theenergy, k = 2 , M , V is the interactionvolume, set to a corresponding interaction radius of r =0 . F BRr is a Breit-Wigner factor given by: F BRr = 1 π M r Γ i ( M r − m r ) + M r Γ r , (4)where M r is the invariant mass of the resonance, witha corresponding decay width of Γ r , and a pole position m r . The product Q r ∈ R goes through all of the resonantparticles, while for stable hadrons only the momentumintegral remains, with F BRr = 1.The last ingredient to the hadronization probabilityis the quark combinatorial factor and the correspondingquark creational probabilities. This is, in some way, simil-iar to the parton model [28,29], where the partonic crosssections are integrated out with the parton density func-tions giving a hadronic cross section. In this model thenumber of quark-antiquark pairs are estimated from sim-ple phase space considerations and is given by [30]: N ( E ) = 1 + p E /T , (5)where T is the hadronization temperature. For each quarkthere is a creational probability P i , i = u, d, s, c, b and thequark distribution function is then given by the multino-mial distribution: F ( N, n i ) = N ( E )! Q i = u,d,s,c n i ! Y i = u,d,s,c P n i i , (6)where n i is the number of quarks of type i = u, d, s, c, b .The distribution function gives the probability of aspecific quark number configuration. In our model, we takethe configuration, where the probability is at maximum,which corresponds to a quark number n i = P i N . Withthe number of quarks, the number of possibilities of a fi-nal state is then calculated with simple combinatorics, andmultiplied by a color factor given by the number of color-less combinations, then normalized by the possible two-, ´abor Balassa, Gy¨orgy Wolf: Charmed and bottomed hadronic cross sections from a statistical model 3 and three body final state factors, giving a final quarkcombinatorial probability. Finally this probability is mul-tiplied by the maximum value of the distribution function,giving the final quark combinatorial factor C Q i . Whenthe expected number of quarks are very small, which isthe case for the heavy quarks with small quark creationalprobabilities, a suppression factor is introduced. This isdescribed in Sec.3, where we fit the charm and bottomquark creational probabilities.If the calculated process respects the conservation lawsand the quarks/antiquarks from the colliding particles arealso included into the combinatorial factors, meaning n i = n i + n initial will be the number of a specific quark/antiquark,there will be no quarks or antiquarks left unpaired at theend, and every quark, which is not used in the final state,could annihilate with a corresponding antiquark.Putting together the previously described factors thenormalized hadronization probability for k fireball is givenby Eq.7. 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 a ) P j T j ( x a ) δ (cid:16) k X a =1 x a − E (cid:17) , (7)where P fbk ( E ) is the fireball decay scheme probability for k -fireballs, N i ,...i k is a symmetry factor counting the fire-balls with the same hadrons after hadronization, x min and x max limits are given by kinematical considerations,and Z k ( E ) is the energy-dependent k-fireball normaliza-tion factor given by: Z k ( E ) = X
15. For the charmand bottom quarks the fit is not that straightforward dueto their considerable masses and therefore their huge sup-pression. Due to the very small charmed and bottomedprobabilities it is assumed that only one charm/bottomquark-antiquark pair is created, but the following argu-ments can be made to more charm and bottom quarksas well. Due to this assumption the constrained proba-bility (one charm/bottom quark is certainly created) ob-tained from the quark number distribution Eq.6, will bemuch smaller than the non-constrained maximal proba-bility. The suppression for both heavy quarks is comingfrom the ratio of the maximal and the constrained proba-bility, obtained from the quark number distribution massfunction, and can be expressed as: γ c,b = F ( N, n ′ u , n ′ d , n ′ s , n ′ c , n ′ b ) F ( N, n u , n d , n s , n c , n b ) = n s ! P c,b ( n s − P s = P c,b N (10)where it is assumed that only one heavy quark is cre-ated, replacing one strange quark, meaning n ′ c,b = 1, and n ′ s = n s −
1, while n ′ u = n u and n ′ d = n d . It is evident thatthe suppression is energy dependent even when the quarkcreational probability is constant and the ratio of theglobal and the constrained maximum tends to unity as theenergy goes higher. After this point the global maximumvalue of the quark number distribution mass function isused and the expected number of heavy quarks, which cor-responds to the global maximum are n c,b = P c,b N , so thetransition between the constrained and the global maxi-mum case is continous.A constant quark creational probability is however nottoo realistic, because at high energies these parameters areessentially have to be equal, therefore we assume a simplelinear relationship between the energy and the quark cre-ational probability P c,b = a c,b E , for the charm and bottomquarks. For the strange quarks, we keep the constant P s value, as it was sufficient to describe exclusive K , Λ , andinclusive strange vector meson production cross sectionsin the energy range of a few tens of GeV’s, where we in-tend to use the model. It is possible that at higher energiesthese values will change, but at this relatively low energyrange the linear relationship for the heavy quarks and theconstant values for the up, down and strange quarks seemto be good approximations. It is also possible that the riseof the strange quark creational probability is so small that G´abor Balassa, Gy¨orgy Wolf: Charmed and bottomed hadronic cross sections from a statistical model a [GeV -1 ] -3 E li m i t [ G e V ] Fig. 1.
Limit energy dependence on the slope parameter a c,b inthe heavy quark creational probability P c,b = a c,b E . The limitenergy marks the energy, where the constrained maximum ofthe quark number distribution function in Eq.6 coincides withits global maximum. P s can be approximated by a constant value at the energyrange considered. Using Eq.10 and the energy dependent P c,b , the heavy quark suppression can be expressed as: γ c,b = a c,b E + E p E /T , (11)where a c,b slope parameters have to be determined fromexperiments. The suppression is energy dependent throughthe number of quarks, and P c,b . This is of course onlyvalid until γ c,b reaches unity, as after that point the globalmaximum will naturally give one heavy quark, and it isnot necessary to constraint the quark number distributionanymore. The actual energy when this happens, dependson the slope parameters a c,b of the charm and bottomquarks and can be calculated setting γ c,b = 1 and solvingthe following equation: a c,b E L + E L p E L /T − , (12)where E L is the limit energy. On Fig.1 the limit energy isshown as the function of the slope parameter a c,b .As the quark creational probability depends only on a c,b , this is the only parameter that has to be fitted fromexperiments. In [22] a very simple fitting procedure isused where, a c was fitted only to one data point from the π − p → J/Ψ X reaction and checked if it gives the correctvalues to different measurement points, in different colli-sions as well. In this paper a least squares fitting methodis used, where an optimum point is sought in the errorfunction, defined as:
Error = vuut N N X i =1 (cid:16) meas i − model i meas i (cid:17) (13) a c [GeV -1 ] -4 E rr o r Fig. 2.
Error function for the inclusive
J/Ψ production. Theminimum corresponds to the slope a c = 8 . · − GeV − . where meas i is a measurement point, model i is the cor-responding model calculation, and N is the number ofmeasurement points used for fitting.For the charm quarks the inclusive π − N → J/Ψ X , and pp → J/Ψ X reactions were used, where the measured val-ues were collected from [31,32,33,34,35,36,37,38,39,40,41,42]. To calculate the inclusive cross section from themodel, apart from the prompt
J/Ψ production, the radia-tive and hadronic decays from the excited charmoniumstates Ψ (3686), χ c , and χ c are included as well, usinga two fireball decay scheme. In Fig.2 the calculated er-ror for the J/Ψ production cross section is shown, wherea clear minimum can be seen at a c = 8 . · − GeV − ,which is really close to our previous fit in [22]. The causeof the difference is that now more particles are introducedin the model e.g. new charmed mesons/baryons, bottomedmesons/baryons, which in exchange slightly changes thenormalization values as well.For the bottom quarks the fitting procedure is thesame as it was for the charm quarks. The process π − p → Υ (1 S ) X is used for fitting, where the measured pointsare taken from [43,44]. In the model calculations apartfrom the direct Υ (1 S ) meson, the excited onium states Υ (2 S ), Υ (3 S ), χ b (1 P ), χ b (1 P ), χ b (2 P ), and χ b (2 P )are used as well. The results for the error function canbe seen in Fig.3, where a minimum is obtained, now at a b = 1 . · − GeV − .The energy dependent quark creational probabilitiesand the suppressions can be seen in Fig.4, where the lim-iting energy - the energy, where the suppression from theconstrained quark number distribution breaks down - isalso shown. After the limiting energy, P c,b still increasesuntil a full equilibrium is reached with the other type ofquarks, however, we do not intend to use the model un-til that energy. The probabilities are energy dependent,therefore the previously fitted P u , P d , and P s values haveto be changed accordingly, to P ′ i = P i − P c / − P b / i = u , d , s . Due to the smallness of the charm and ´abor Balassa, Gy¨orgy Wolf: Charmed and bottomed hadronic cross sections from a statistical model 5 a b [GeV -1 ] -5 E rr o r Fig. 3.
Error function for the inclusive Υ (1 S ) production. Theminimum corresponds to the slope a b = 1 . · − GeV − . E [GeV] P c E L = 19.32 GeVP c = (8.5 10 -4 GeV -1 ) E
20 40 60 80 100 120 140 160 180 200
E [GeV] P b -3 E L = 174 GeVP b = (1.05 10 -5 GeV -1 ) E Fig. 4.
The energy depenedence of the quark creational prob-abilities and the limiting energies for the charm (upper panel)and bottom (bottom panel) quarks. bottom probabilities, the corrections are very small at theenergy range we are interested in, so in practical calcula-tions P ′ i ≈ P i can be used.In the next section the results for different onium stateproduction, as well as some open charm calculations areshown for proton-proton, pion-proton, and proton-antiprotoncollisions. After the quark creational probabilities for heavy quarksare obtained, it is possible to calculate inclusive cross sec-tions for charmed and bottomed hadrons as well. The firstreaction shown is the energy dependence of the pp → J/Ψ X process, which is used to fit the charm quark cre-ational probability. The decays from the excited Ψ (3686), E [GeV] -1 p p J / X [ nb ] meas.modelmodel error Fig. 5.
Model calculation for the pp → J/ΨX inclusive crosssection. The black and red lines are the model result and its er-ror respectively, while the circles are measurement points from[31,32,33,34,35,36,37,38,39,40,41,42]. χ c , χ c states are also included in the calculations. Theresults can be seen in Fig.5, where as expected, the agree-ment with the data is really good. The second exampleis the π − p → J/Ψ X process, which is expected to givea few times larger cross section compared to the proton-proton case. The results can be seen in Fig.6, where againthe agreement is still satisfactory. The third example isthe π − p → Υ (1 S ) X process, where apart from the di-rect bottomonium production, the decays from the excitedonium states Υ (2 S ), Υ (3 S ), χ b (1 P ), χ b (1 P ), χ b (2 P ),and χ b (2 P ) are included. The results can be seen in Fig.7.The agreement between measurement and model in theprevious three processes are, however expected as thesewere the ones used to fit the quark creational probabili-ties.The fourth process is the pp → Υ X , where Υ = Υ (1 S )+ Υ (2 S ) + Υ (3 S ). The measured data points are at √ s =29 . √ s = 29 . pp → J/Ψ , whichonly gives a significant contribution near threshold. Theresults can be seen in Fig.9. The proton-antiproton col-lision is therefore favorable in constrast to the proton-proton case, however its production rate is still lower thanin pion-proton collisions. This is not just a curiosity on itsown, because in heavy ion collisions an interesting prob-lem is the propagation of onium states in dense nuclearmatter, as they could have a significant mass shift, which
G´abor Balassa, Gy¨orgy Wolf: Charmed and bottomed hadronic cross sections from a statistical model
E [GeV] -1 - p J / X [ nb ] meas.modelmodel error Fig. 6.
Model calculation for the π − p → J/ΨX inclusive crosssection. The black and red lines are the model result and itserror, while the circles are measurement points from [31,32,33,34,35,36,37,38,39,40,41,42].
12 14 16 18 20 22 24 26 28 30
E [GeV] - p X [ pb ] meas.modelmodel error Fig. 7.
Model calculation for the π − p → Υ X inclusive crosssection. The black and red lines are the model result and its er-ror respectively, while the circles are measurement points from[43,44]. is proportional to the gluon condensate [47,48], so if onecould measure the mass shift of e.g. Ψ (3686) the nuclearexpectation value of the gluon condensate at a specific nu-clear density can be deduced. Due to the really small crosssections it is necessary to create the most onium states onecan get, which is one reason to use antiprotons or pionsin these investigations.Some of our transport calculations[49,50] requires thedetermination of the direct charmonium production, with-out the decays from higher states as well. As there are onlyjust a few measured values for e.g. direct Ψ (3686) produc-tion, we checked the ratios defined in Eq.14 and in Eq.15 Index of meas. point at 29.1 GeV CM energy pp X [ pb ] datamodelmodel error
12 14 16 18 20 22 24 26 28 30
E [GeV] -2 pp X [ pb ] Fig. 8.
Model calculation for the pp → Υ X inclusive crosssection. The black and red lines are the model result and itserror, while the circles and crosses are measurement pointsfrom [45].
E [GeV] p p J / X / p pba r J / X
10 12 14 16 18 20 22 24 26 28 30
E [GeV] p p X / p pba r X Fig. 9.
Inclusive charmonium (upper panel) and bottomo-nium (lower panel) cross section suppression ratios in proton-proton to proton-antiproton collisions. The pp → J/Ψ and pp → Υ (1 S ) one body final state Breit-Wigner cross sectionsare not included in the model calculations. The measured valuefor charmonium production is taken from [46]. for the χ c , χ c and Ψ (3686) particles: R ABχ c + χ c = X i =1 Br( χ ci → J/Ψ ) σ AB → χ ci X σ AB → J/ΨX (14) R ABΨ (3686) = Br( Ψ (3686) → J/Ψ ) σ AB → Ψ (3686) X σ AB → J/ΨX , (15)where A and B represents the colliding particles, in thiscase π − and p , and Br is the branching fraction of theparticles to decay into a J/Ψ meson. The measured ratios ´abor Balassa, Gy¨orgy Wolf: Charmed and bottomed hadronic cross sections from a statistical model 7 R pNΨ (2 S ) R πNΨ (2 S ) √ s [GeV] 23 . . . . . ± . ± . ± . . ± . . ± . ± . . ± . Table 1.
Total
J/Ψ production ratio to the direct Ψ (3686)production cross section. The measured values are collectedfrom [51,52,53,54,55,56]. R πNχ c + χ c √ s [GeV] 18 . . . . ± . ±
14 34 ± . ± . . ± . . ± . Table 2.
Total
J/Ψ production ratio to the direct χ c , χ c production cross sections. The measured values are collectedfrom [51,52,53,54,55,56]. are collected from [51,52,53,54,55,56] and the comparisonwith the model calculations at different collision energiesare shown in Table.1 and in Table.2, where a really goodmatch with the data is achieved. This allows us to calcu-late the full energy dependence of the direct productionof these states and put it into our transport simulations.The following few processes are somewhat more com-plicated than the previous ones. These are the inclusiveopen charm productions in π − p and in pp collisions. Themain problem is that our model is heavily based on theknown resonances, excited states, which could decay intohadrons we are interested in, and if the branching fractionsor the quantum numbers of such particles are not wellknown, then it could not be included in the calculations.The problem with open charm (and open bottom) crosssections is therefore lies in the fact, that there are manyexcited mesons and baryons that could decay into groundstate open charm/bottom mesons, where the branchingfractions are not known at all. Nevertheless, we still try togive an order of magnitude estimation to these cross sec-tions, namely to: πp → D + /D − X , πp → D /D X , andthe pp → D + /D − X , pp → D /D X . To all of these pro-cesses the resonances D ∗ (2007) , D ∗ (2010) ± , D ∗ (2300) , D ∗ (2300) ± , D (2420) , D ∗ (2460) , D ∗ (2460) ± , D ∗ (2750), Λ c (2860) + , Λ c (2880) + , Λ c (2940) + are included, whose quan-tum numbers are at least fairly well known in the PDG.However, the branching fractions are not known at all, andin the excited Λ c baryon resonances even less is knownthan the D mesons. To be able to give an estimation, weintroduced an asymmetry parameter r defined as: r = Br( X → D + /D − )Br( X → D /D ) , (16)with the assumption that every other decay is negligi-ble, so Br( X → D + /D − ) + Br( X → D /D ) ≈ X ’ Fig. 10.
Quark flow diagrams to the decays Λ + c → D + n and Λ + c → D p . e.g. D ∗ (2300) giving a D + to the expense of D , andvaried it between [0 . , i ∈ [0 . , . D ∗ (2007) + decays, where the branching frac-tions are known, the asymmetry in the decay productsis r ≈ .
46. For the Λ resonances, where the final stateproducts are also not well known, we assumed it could de-cay as Λ + c → pD , Λ + c → nD + , and for the antiparticles Λ c → pD , Λ c → nD − according to the quark flow dia-grams in Fig.10, with the same asymmetry variations asin the meson resonances. In [57] an effective model calcu-lation estimated the asymmetry to r ≈
1, and it is highlypossible that each resonance has different branching frac-tions, so the best we can do at this point is to stay atthe varied asymetries and give an order of magnitude es-timation. It could be an interesting further step to includeall the possible resonances and branching ratios from e.g.effective model calculations. Using the mentioned assump-tions, the results are shown in Fig.11-Fig.14, where onlycross section intervals are shown due to the varied asym-metries. It can be seen that an order of magnitude esti-mation is very much achievable, even with the high uncer-tainty in the parameters.
We extended our model with the inclusion of charmedand bottomed hadrons and estimated the charm and bot-tom quark creational probabilities, assuming a simple lin-ear relationship between the probability and the energy P c,b = a c,b E . A fit is made for the two slope parameters G´abor Balassa, Gy¨orgy Wolf: Charmed and bottomed hadronic cross sections from a statistical model
Fig. 11.
Estimated cross section interval for the π − p → D + /D − process, with varied branching fraction asymetries.The data is taken from [58,59,60]. Fig. 12.
Estimated cross section interval for the π − p → D /D process, with varied branching fraction asymetries. Thedata is taken from [58,59,60]. a c , and a b , using inclusive charmonium and bottomoniumdata in proton-proton and in pion-proton collisions. Thesuppression of the higher charmonium states Ψ (3686), χ c ,and χ c to the direct J/Ψ production is also calculatedand compared to the measured ratios in π − p and pp colli-sions, giving a really good match with the data. The modelis further validated through the processes pp → J/Ψ X , π − p → J/Ψ X , pp → Υ X , and π − p → Υ X . Estimationswere made to proton-antiproton collisions and the ratios σ pp → J/ΨX /σ pp → J/ΨX , σ pp → Υ X /σ pp → Υ X are calculated. Itcan be concluded that at low energies proton-antiprotonand pion-proton collisions are very much favorable if onewants to produce charmonium particles. For further vali-dation open charm production cross sections were also es-timated, namely the π − p → D + /D − X , pp → D + /D − X , π − p → D /D X , and the pp → D /D X processes. Fig. 13.
Estimated cross section interval for the pp → D + /D − process, with varied branching fraction asymetries. The datais taken from [58,59,60]. Fig. 14.
Estimated cross section interval for the pp → D /D process, with varied branching fraction asymetries. The datais taken from [58,59,60]. The main problem with these calculations are the miss-ing branching fractions in the PDG, which are neededto make reliable estimates, however introducing an asym-metry parameter, an order of magnitude estimation wasmade, which is in agreement with the measurements. Withthese extensions the model is now considered complete re-garding non-exotic states. It is however a further ques-tion how to include e.g. tetra-quarks into the model, asthere are no measurements for these states at low ener-gies, where the model works.
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