Phenomenology of strangeness production at high energies

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by  Abdel Nasser Tawfik

epl draft
Phenomenology of strangeness production at high energies
Abdel Nasser Tawfik1,2, Hayam Yassin3, Eman R. Abo Elyazeed3, Muhammad Maher2,4, Abdel Magied
Diab1,2, Magda Abdel Wahab3 and Eiman Abou El Dahab1,2,5
1 Egyptian Center for Theoretical Physics (ECTP), MTI University, 11571 Cairo, Egypt.
2 World Laboratory for Cosmology And Particle Physics (WLCAPP), 11571 Cairo, Egypt.
3 Physics Department, Faculty of Women for Arts, Science and Education, Ain Shams University, 11577 Cairo,
Egypt.
74 Physics Department, Faculty of Science, Helwan University, 11795 Cairo, Egypt.
1
5 Faculty of Computer Sciences, MTI University, 11671 Cairo, Egypt
0
2

n
a PACS 25.75.-q–Relativistic heavy-ionnuclear reactions
J
PACS 25.75.Dw–Particle production (relativistic collisions)

6 PACS 24.85.+p–Quantumchromodynamics in nuclei
1
Abstract – The strange-quark occupation factor (γs) is determined from the statistical fit of
h] themultiplicityratioK+/π+ inawiderangeofnucleon-nucleoncenter-of-massenergies(√sNN).
p Fromthissingle-strange-quark-subsystem,γs(√sNN)wasparametrized asadampedtrigonomet-
- ric functionality and successfully implemented to the hadron resonance gas model, at chemical
p semi-equilibrium. VariousparticleratiosincludingK−/π−,Λ/π−,andΛ¯/π− arewellreproduced.
e
Thephenomenologyofγs(√sNN)suggeststhat,thehadrons(γs raises) at√sNN 7GeVseems
h ≃
[ to undergo a phase transition to a mixed phase (γs declines), which is then derived into partons
(γs remains unchanged with increasing √sNN), at √sNN 20 GeV.
≃
1
v
6
1
8
4 Introduction . – The energy scan of the Super Pro- ture of K+/π+, we first observe an increase in K+/π+
0
tonsynchrotron (SPS) confirming continuation and com- withincreasingthenucleon-nucleoncenter-of-massenergy
.
1pletion of the set of excitation functions observed at the (√s ) up to a certain value (at projectile energy of
NN
0
AlternatingGradiantSynchrotron(AGS)-amongothers- 30 AGeV [12]), which is then followed by a rapid de-
17theratiosofvariousparticlemultiplicities. Widely-known ∼crease. At higher √sNN, K+/π+ remains almost un-
:examplesincludetheso-calledMarek’shorn-likestructure changed shaping a plateau at top RHIC energies. This
vof K+/π+ [1], which was confirmed in various heavy-ion continues up the Large hadron Collider (LHC) energies,
i
Xcollisionsexperiments[2–4],thekick,whichistobedrawn where another puzzle was reported, recently, namely the
rthrough a rapid change in the slope of thr produced pi- proton anomaly or the discrepancy between calculated
aons per participating nucleon and the plateau of the aver- and measured p/π ratio [13]. It was conjectured that,
aged transverse momentum p of kaons, known as step. this energy-dependence can be generatedby the hadronic
T
h i
The recent energy scan of the Relativistic Heavy Ion Col- kinetic model [14] and also when assuming a transition
lider (RHIC) confirms the horn-like structure and both from a baryon-dominated system (at low) to a meson-
kick and step observations [3], as well. These phenomena dominatedsystem(athighcollisionenergy)[5]. Introduc-
apparentlyindicateremarkablechangestakingplaceinthe ing heavy masses of unknown hadronic resonances (Hage-
strongly interacting system at relativistic energies. Vari- dorn mass spectrum) [7] is believed to assure a release of
ous explanationswere proposed,so far [1,5–11], including additional color degrees-of-freedom (dof) [1] so that large
the onset of deconfinement, the manifestation of critical numbers of pions even greater than that of the kaons can
endpoint, etc. That the thermal models seem not being beguaranteed. Thelatter-inturn-canbeachievedwhen
able to reproduce such a remarkable structure represents including heavy hadron resonances from the recent com-
one ofthe chronic puzzles facing the particle scientists, to pilationoftheParticleDataGroup(PDG)[6]intheHRG
which a great number of experiments and theoretical and model. All these proposals were implemented but unfor-
numerical studies were devoted. tunately not able to give an unambiguous clarification,so
When focusing the discussion of the horn-like struc- far.
p-1
A. Tawfik et al.
Fortheproductionandevolutionofstrangequarks(and Approach. – In the partition function of a grand-
strange hadrons) with varying collision energies, a micro- canonical ensemble describing a strongly interacting sys-
scopicmodelwasalternativelyutilized[15],wherethemo- tem, statistically, γ and γ can be integrated in as pre-
l s
mentum integrated Boltzmann equation should be first factors to the Boltzmann exponential,
evaluated. In the Bag model, the ratios of strangeness
∞
Vg
to entropy was studied in dependence on the collision en- lnZ(T,µ) = i k2dk
ergy √sNN [16]. Interestingly, a scaling behavior similar ±Xi 2π2 Z0
twohitchhecoonnesidfreorms thheadernoenrigcy-ndoenpeeqnudiliinbgriluifmetikmineeotficthmeofidreel-, ln(cid:20)1±(γlnl)i (γsns)i e(cid:16)µi−Tεi(k)(cid:17)(cid:21), (2)
ball [9] was obtained. Both are found similar to the ob-
served K+/π+. The present letter introduces an energy- where standsforfermionsandbosons,respectively,and
dependent strange-quark occupation factor (γ ) deduced, ε (k) =± (k2 + m2)1/2 is the energy-momentum relation
s i i
phenomenologically, and assures a best reproduction of of the i-th hadron. Implementing Hagerdon picture that
various particle ratios. heavy resonances are composed of lighter ones which in
The occupation factors of light- and strange-quarks, γ tern are consisting of ligher ones and so on, the subscript
l
andγ ,respectively,werefirstintroducedbyLetessierand i refers to a summation over fermions and bosons from
s
Rafelskiasa plausible explanationfor the strangenessen- recent compilation of PDG. The chemical potential µi is
hancement [17], which was proposed as a sensitive sig- composed of various contributions, for instance,
nature for the formation of quark-gluon plasma (QGP).
µ =B µ +S µ + , (3)
It was argued that the energy threshold for s¯s pair- i i B i S ···
production considerably differs due to the underlying
where B and S are baryon and strangeness quantum
i i
QCD symmetries. In QGP phase, ǫthr 300 MeV, while number of i-th hadron resonance and µ and µ are the
≃ B S
in the hadron phase, ǫthr 700 MeV, because the energy baryon and strangeness chemical potential, respectively.
≃
threshold corresponds to two times the rest mass of par-
V and T are the fireballvolume and temperature, respec-
tons and hadrons of interest. Thus, it was conjectured
tively.
that the creation of QGP should be accompanied by an
In Eq. (2), n and n are the number of light- and
l s
increaseinthestrangenessproduction. Thiswasfirstcon-
strange-quarks, respectively, of which each fermion or
firmed, experimentally [18] and also in the first-principle
boson is composed. When unity is assigned to γ and
l
lattice QCD simulations [19]. Also, it was assumed that
γ , full chemical equilibrium, as the case in the equilib-
s
the nonequilibrium values, i.e. the ones differing from
rium hadron resonance gas (HRG) model, the calculated
unity,assignedtoγs wouldbeabletocharacterizetheleg- K+/π+ poorlygeneratesthehorn-likestructuremeasured
end energy-dependence of K+/π+ [10]. The present work
at SPS energies, but greatly overestimates the results at
isanextensionandupdatingofRef.[10]withmostrecent
high√sNN. ItisnoteworthyhighlightingthatatAGSen-
experimentalresultsandmoreplausiblephenomenological
ergies,the equilibrium HRG calculations seem to describe
explanations.
well the production rates of strange hadrons [10,11,22].
From the experimental point-of-view, the relative pro- In this energy-limit, the HRG calculations seem not sen-
duction of strange and nonstrange quarks was analysed sitive to γ as to the resonance masses and the excluded
s
in elementary and in nucleus collisions [20]. Relative the volume corrections, etc., Fig. 2. The statistical nature of
the light-quark pair production, it was concluded that s¯s strangeness production, in our case kaon, is conjectured
pairs would be considerably suppressed. Through quark to be maintained, when ad hoc nonequilibrium values are
counting, one assumes that u¯u : d¯d : s¯s = 1 : 1 : λs, assigned to the strange-quark occupation factor (γs = 1)
6
where λs γs 1 apart from some factors characteriz- [17]. The present letter concludes a rapid strangeness en-
≡ ≪
ingthe resultsfromheavy-ioncollisions. Thisobservation hancementatAGSenergies,atequilibriumlight-quarkoc-
wasfirstconfirmedinheavy-ioncollisionsat√s 60GeV cupation factor (γ =1), i.e. chemical semi-equilibrium.
≃ l
[21]. It was reportedthat the experimentalestimationfor A strangeness suppression factor, i.e. γ < 1, was ex-
s
averagemultiplicities of Ns¯s allowedthe conclusionthat plicitly assumed in Ref. [23],
h i
λ isnotsensitivetotheinteractingsystem,e.g. quantum
s
numbersofcollidingnuclei,butapparentlytothecollision
γ (√s ) = 1 α exp β A√s , (4)
s NN NN
energies [21]. − (cid:18)− q (cid:19)
whereα=0.606,β =0.0209,andAistheatomicnumbers
λ = hNs¯si , (1) of the colliding nuclei. This has the advantage to reduce
s
Nq¯q Ns¯s the production rate of the strange hadrons [23].
h i−h i
In the present work,we also assume γ =1. But for γ ,
l s
wdehteerremhinNeq¯dqifr=omhNthu¯euide+tehcNtido¯dnio+f shtNras¯snig,ei.mees.onλssacnadntbhee wcaenfibteoduerrivHeRdGfrocmalcEuqla.t(io2n),s,onwKhe+re/πt+heatnuvamrbyeinrgd√enssNitNy
total meson multiplicity. with the experimental results, top panel of Fig. 1. From
p-2
Phenomenology of strangeness production at high energies
the statistical fit to this single-strange-quark-subsystem, within the energy range 7.√s .20 GeV, γ rapidly
NN s
we aim to parametrize γ as functionality of √s . De- decreases. At higher energies, γ becomes energy inde-
s NN s
tailsaboutHRGcanbetakenfromRef.[24],where√sNN pendent, especially at top RHIC and LHC energies. Its
is related to baryon chemical potential (µ ) and thus di- asymptotic value is given by the parameter f in Eq. (5).
B
rectly enters the partition function, Eq. (2). For specific
particle species, their number densities are composed of
the contributions coming from the correspondinghadrons 1.2
and that stemming from heavy resonances decaying into
1
that particles of interest. The latter should be weighted
by the corresponding branching ratios.
0.8
Within the extensive statistics, it is conjectured that, γs
the colliding system reaches the stage of chemical freeze- 0.6
out, which is characterized by two thermodynamic quan- γs from K+/π+ fit
0.4 proposed expression
tities; T and µB, which can be deduced from statisti- IJMPE25,1650058(2016)
cal fits of the HRG calculations of various particle ratios
0.2
and/oryieldstotheexperimentalmultiplicities. Resulting 1 10 100 1000 10000
T and µ are well described by various freezeout condi- √sNN [GeV]
B
tions[24–27]. Concretely,weimplements/T3 =7[28,29],
Fig.1: Strange-quarkoccupationfactor, γs,isgivenasafunc-
with s being the entropy density. In fitting the experi-
tion of √sNN. Symbols with errors depict our results from
mental K+/π+ to the HRG calculations, we assume γs as the statistical fits of measured K+/π+ to the HRG calcula-
a free parameter, while the values of the parameters T tions with γs taken as a free parameter. The dashed curve
and µB are determined at s/T3 =7 and γq is assigned to represents Eq. (4).
a constant value. As mentioned, we concentrate the dis-
cussiononfittingwiththesingle-strange-quark-subsystem The top panel of Fig. 2 shows K+/π+ (symbols) as a
K+/π+ andassumethattheresultingγ (K+/π+)remains functionof√s ,whichinturnisrelatedtoµ . Dashed
s NN B
valid to the entire HRG thermal model. curve gives HRG calculations, at equilibrium occupation
parameters,namelyfullchemicalequilibriumγ =γ =1.
Results.– InFig. 1,theresultingγ aredepictedasa l s
s The double-dotted curve illustrates results at γ = 1,
function of √s (symbols with errors). At the chemical l
NN while γ is determined from Eq. (4) [17]. The earlier
freezeout, these are well described by a damped trigono- s
estimations excellently reproduce the K+/π+ results at
metric functionality,
√s .10 GeV. At higher energies, although they over-
NN
γ (√s ) = a exp( b√s ) sin(c√s +d) estimatetheexperimentalresults,ahorn-likestructurere-
s NN NN NN
−
mains guessable. The latter calculations don’t posses any
+ f, (5)
horn-like structure. At high energies, they fit well with
witha=2.071 0.259,b=0.282 0.062,c=0.362 0.051, the results from equilibrium HRG, i.e. overestimate the
± ± ±
d=4.78 0.21,andf =0.764 0.249. Expression(5)isil- measurements, as well.
± ±
lustratedasthe solidcurve. For the sakeofcompleteness, Our results are presented as the solid curve. They are
wecompareourresultswithpredictionsdeducedfromEq. deduced from ideal HRG calculations, i.e. point-like con-
(4), which is given as the dashed curve. So far, both ex- stituents, PDG compilation, etc. In these calculations,
pressions do not allow for any concrete conclusion. This chemical semi-equilibrium, namely γl(√sNN) = 1 but
might be only possible, when they are (or not) succeeded γ (√s ) is to be determined from Eq. (5) are assumed.
s NN
in reproducing other particle ratios as illustrated in Fig. FromthefactthatEq. (5)isstemmingfromthestatistical
2. fit of the HRG calculations to the measured K+/π+, the
When the energy-dependence of γ , Eq. (4), [23] is im- excellent agreement between the solid curve and K+/π+
s
plemented, wefirstfind thatthe resulting γ is monotoni- results is very obvious. To judge about the validity of
s
cally depending on √sNN. At √sNN .200 GeV, we find our parametrization, Eq. (5), we still need to utilize it
that, γ exponentially increases, while at higher energies, in calculating other quantities and we might also need to
s
√sNN likely approaches an asymptotic value; the chemi- run further checks, such as, lattice QCD thermodynam-
calequilibrium. Atγ =1andEq. (4),fourparticleratios icsandexaminethethermodynamicconsistency,etc. The
l
are calculated and shown on Fig. 4. This shall be elab- latter shall be subjects of future studies. For the phe-
orated, shortly. For now, we merely highlight that, the nomenological focus of the present letter, we concentrate
results are almost identical to the ones calculated at full on calculating various particle ratios.
− − −
chemical equilibrium. Accordingly, one can easily judge Panelsb),c),andd)ofFig. 2depictK /π ,Λ/π ,and
about the improvement made through Eq. (4). Λ¯/π−, respectively, as functions of √sNN. These experi-
Our parametrization for γ , Eq. (5), contrarily shows mental results are confronted to the HRG calculations at
s
a nonmonotonic energy-dependence. At √sNN . 7 GeV, full chemicalequilibrium, i.e. γl(√sNN)=γs(√sNN)=1
γ exponentially increases with increasing √s . Then, (dashedcurves)andquasi-equilibrium,i.e. γ (√s )=1
s NN l NN
p-3
A. Tawfik et al.
0.3 experimental results is remarkably improved. Thus, we
(a)
conclude that our parametrization for γ (√s ) enables
0.25 s NN
the thermal models, the HRG for instance, to reproduce
0.2 variousparticleratios. RelativetoEq. (4), Eq. (5)agrees
++πK/ 0.15 very well with the experimental results.
0.1 Discussion and Conclusions. – AsdiscussedinIn-
troduction,variousinterpretationsforthehorn-like struc-
0.05
ture measured in K+/π+ have been proposed, for exam-
0
1 10 100 1000 ple, parametrization for the evolution of the fireball and
√sNN [GeV] theproduced-particledensities[9]. Itwasnoticedthat,for
beam energies >30 AGeV, the fireball lifetime decreases,
0.2 (b) whoseinterplaywiththeinitialenergydensitydetermines
the final-state density. The hadronic kinetic model based
0.15 ontheseassumptionsbesidesfullchemicalequilibriumare
--πK/ 0.1 cinognjKec+tu/πre+d.todescribewelldifferentparticleratiosinclud-
IJMPE25,1650058(2016)
This work
HRG, γs=1 Also, to interpret the energy-dependence of the kaon
0.05 AGS
SPS production,amicroscopicapproachandfullchemicalequi-
RHIC
LHC
librium have been proposed [15]. Here, the evolution of
0
1 10 100 1000 strangenessproduction was modelled by momentum inte-
√sNN [GeV]
gratedBoltzmannequation. This approachwas borrowed
0.25 from the freezeout processes in the thermal expansion of
(c)
the early universe [30]. When an initial partonic phase is
0.2
assumedatcollisionenergiesgreaterthanacertainthresh-
0.15 old, a non-monotonic energy-dependence of K+/π+ was
-Λπ/ obtained.
0.1
The excitation functions in the particle multiplicities
K+/π+ observed at SPS energies and confirmed in the
0.05
RHIC energy-scan, especially the horn-like structure, are
0 analysed in the HRG statistical-thermal model, in which
1 10 100 1000
√sNN [GeV] point-like constituents and chemical semi-equilibrium are
considered. The strange-quark occupation factor is as-
0.05
sumed to vary with the collision energy. This idea was
(d)
0.04 introduced and worked out by many authors, including
AT. Additional to a best reproduction of various parti-
-Λπ/ 0.03 cle ratios, the present work introduces an updating with
anti 0.02 most recent experimental results and presents a novel
parametrization of the dependence of γ on the colli-
s
0.01 sion energies. The resulting functionality describing γs
with varying √s is proposed as damped trigonomet-
NN
0 ric functionality. Accordingly, such a monotonic energy-
1 10 100 1000
√sNN [GeV] dependence of γ was successfully implemented to the
s
HRG model at chemical semi-equilibrium. As discussed,
Fig.2: Theexperimentalresultsofdifferentparticle-ratios are the particle ratios K−/π−, Λ/π−, and Λ¯/π− are well re-
given in dependence on √sNN. The solid curves depict the
produced.
present calculations at γl = 1 and γs is determined from Eq.
Inlightofthis phenomenologicwelldescriptionandthe
(5). Double-dashedanddashed curvesrepresent idealHRGat
assumption that the light-quark occupation factor is as-
full chemical equilibrium (γl = γs = 1) and semi-equilibrium
[γl =1 and γs is given byEq. (4)], respectively. signed to the equilibrium value, we can conclude that
γ . 1, i.e. chemical semi-equilibrium, at all collision en-
s
ergies. Only at low SPS energy, one obtains that γ 1
s
→
and γs(√sNN) = 1. The latter is given by Eq. (4) [17] andevenslightlyexceedsthischaracteristicvaluederiving
(double-dottedcurves). Wefindthattheyarealmostcom- the colliding system to full chemical equilibrium. Accord-
patible with the HRG calculations at full chemical equi- ingly, we expect that such a state should possess maxi-
librium. mumentropy. Although,otheraspectsonthermodynamic
OurHRGcalculationswithEq. (5)arepresentedasthe consistency shall be analysed in future works, we merely
solid curves. Here, we find that, the agreement with the highlight that an old paper of AT [10] reported on the
p-4
Phenomenology of strangeness production at high energies
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