research article hawking-unruh hadronization and ...research article hawking-unruh hadronization and...

Research ArticleHawking-Unruh Hadronization and StrangenessProduction in High Energy Collisions

Paolo Castorina123 and Helmut Satz4

1 Dipartimento di Fisica ed Astronomia Universita di Catania Via Santa Sofia 64 95100 Catania Italy2 INFN Sezione di Catania Via Santa Sofia 64 95100 Catania Italy3 PH Department TH Unit CERN 1211 Geneva 23 Switzerland4 Fakultat fur Physik Universitat Bielefeld Germany

Correspondence should be addressed to Paolo Castorina paolocastorinactinfnit

Received 8 January 2014 Accepted 16 April 2014 Published 11 May 2014

Academic Editor Piero Nicolini

Copyright copy 2014 P Castorina and H Satz This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

The thermal multihadron production observed in different high energy collisions poses many basic problems why do evenelementary 119890+119890minus and hadron-hadron collisions show thermal behaviourWhy is there in such interactions a suppression of strangeparticle production Why does the strangeness suppression almost disappear in relativistic heavy ion collisions Why in thesecollisions is the thermalization time less than ≃ 05 fmc We show that the recently proposed mechanism of thermal hadronproduction throughHawking-Unruh radiation can naturally answer the previous questions Indeed the interpretation of quark (q)-antiquark (119902) pairs production by the sequential string breaking as tunneling through the event horizon of colour confinementleads to thermal behavior with a universal temperature 119879 ≃ 170Mev related to the quark acceleration a by 119879 = 1198862120587 Theresulting temperature depends on the quark mass and then on the content of the produced hadrons causing a deviation from fullequilibrium and hence a suppression of strange particle production in elementary collisions In nucleus-nucleus collisions wherethe quark density is much bigger one has to introduce an average temperature (acceleration) which dilutes the quark mass effectand the strangeness suppression almost disappears

1 Introduction

Hadron production in high energy collisions shows remark-ably universal thermal features In 119890+119890minus annihilation [1ndash3]in 119901119901 in 119901119901 [4] and more general ℎℎ interactions [3] aswell as in the collisions of heavy nuclei [5ndash11] over an energyrange from around 10GeV up to the TeV range the relativeabundances of the produced hadrons appear to be those of anideal hadronic resonance gas at a quite universal temperature119879119867asymp 160ndash170MeV [12]There is however one important nonequilibrium effect

observed the production of strange hadrons in elementarycollisions is suppressed relative to an overall equilibriumThis is usually taken into account phenomenologically byintroducing an overall strangeness suppression factor 120574

119904lt 1

[13] which reduces the predicted abundances by 120574119904 1205742

119904 and 1205743

119904

for hadrons containing one two or three strange quarks (orantiquarks) respectively In high energy heavy ion collisionsstrangeness suppression becomes less and disappears at highenergies [14 15]

The origin of the observed thermal behaviour has beenan enigma for many years and there is a still ongoing debateabout the interpretation of these results [16ndash24] Indeed inhigh energy heavy ion collisions multiple parton scatteringcould lead to kinetic thermalization but 119890+119890minus or elementaryhadron interactions do not readily allow such a description

The universality of the observed temperatures on theother hand suggests a common origin for all high energycollisions and it was recently proposed [25] that thermalhadron production is the QCD counterpart of Hawking-Unruh (H-U) radiation [26 27] emitted at the event horizon

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 376982 7 pageshttpdxdoiorg1011552014376982

2 Advances in High Energy Physics

due to colour confinement In the case of approximatelymassless quarks the resulting hadronization temperature isdetermined by the string tension 120590 with 119879 ≃ radic1205902120587 ≃

170Mev [25] Moreover in [28] it has been shown thatstrangeness suppression in elementary collisions naturallyoccurs in this framework without requiring a specific sup-pression factorThe crucial role here is played by the nonneg-ligible strange quark mass which modifies the emission tem-perature for such quarks in the Hawking-Unruh approachthe temperature associated with strange particle productionis different from that one of nonstrange particles Althoughthis conclusion has been obtained by a full analysis in thestatistical hadronization model in [28] here we propose anelementary scheme of understanding strangeness suppres-sionwhichmoreover opens the possibility of explaining whythis suppression disappears in heavy ion collisions where thelarge parton density in causally connected space-time regionof hadronization produces a unique temperature for strangeand nonstrange particles

In particular in the second section after recalling thestatistical hadronizationmodel one discusses the subsequentdescription in terms of H-U radiation in QCD includingthe dependence of the radiation temperature on the massof the produced quark and the strangeness suppression inelementary collisions In Section 3 the dynamical mechanismfor strangeness enhancement (ie no suppression) in rela-tivistic heavy ion scatterings is introduced giving a coarsegrain evaluation of the Wroblewski factor [29] a parameterwhich counts the relative number of strange and nonstrangequarks and antiquarks Section 4 is devoted to comments andconclusions

2 Hawking-Unruh Hadronization

In this section we will recall the essentials of the statisticalhadronization model and of the Hawking-Unruh analysis ofthe string breaking mechanism For a detailed descriptionssee [3 25 28]

21 Statistical Hadronization Model The statistical hadr-onization model assumes that hadronization in high energycollisions is a universal process proceeding through theformation ofmultiple colourlessmassive clusters (or fireballs)of finite spacial extension These clusters are taken to decayinto hadrons according to a purely statistical law everymultihadron state of the cluster phase space defined by itsmass volume and charges is equally probable The massdistribution and the distribution of charges (electric bary-onic and strange) among the clusters and their (fluctuating)number are determined in the prior dynamical stage of theprocess Once these distributions are known each cluster canbe hadronized on the basis of statistical equilibrium leadingto the calculation of averages in themicrocanonical ensembleenforcing the exact conservation of energy and charges ofeach cluster

Hence in principle one would need the mentioneddynamical information in order to make definite quantitativepredictions to be compared with data Nevertheless for

Lorentz-invariant quantities such as multiplicities one canintroduce a simplifying assumption and thereby obtain asimple analytical expression in terms of a temperature Thekey point is to assume that the distribution of masses andcharges among clusters is again purely statistical [3] so thatas far as the calculation of multiplicities is concerned theset of clusters becomes equivalent on average to a largecluster (equivalent global cluster) whose volume is the sumof proper cluster volumes and whose charge is the sum ofcluster charges (and thus the conserved charge of the initialcolliding system) In such a global averaging process theequivalent cluster generally turns out to be large enough inmass and volume so that the canonical ensemble becomesa good approximation In other words a temperature canbe introduced which replaces the a priorimore fundamentaldescription in terms of an energy density

To obtain a simple expression for our further discussionwe neglect for the moment an aspect which is important inany actual analysis Although in elementary collisions theconservation of the various discrete abelian charges (electriccharge baryon number strangeness heavy flavour) has tobe taken into account exactly [30] we here consider forthe moment a grand-canonical picture We also assumeBoltzmann distributions for all hadrons The multiplicity ofa given scalar hadronic species 119895 then becomes

⟨119899119895⟩primary

=

1198811198791198982

119895

21205872120574

119899119895

119904 1198702(

119898119895

119879

)(1)

with 119898119895denoting its mass and 119899

119895the number of strange

quarksantiquarks it contains Here primary indicates thatit gives the number at the hadronisation point prior to allsubsequent resonance decay The Hankel function 119870

2(119909)

with 119870(119909) sim expminus119909 for large 119909 gives the Boltzmannfactor while119881 denotes the overall equivalent cluster volumeIn other words in an analysis of 4120587 data of elementarycollisions 119881 is the sum of the all cluster volumes at alldifferent rapidities It thus scales with the overall multiplicityand hence increases with collision energy A fit of produc-tion data based on the statistical hadronisation model thusinvolves three parameters the hadronisation temperature 119879the strangeness suppression factor 120574

119904 and the equivalent

global cluster volume 119881As previously discussed at high energy the temperature

turns out to be independent of the initial configurationand this result calls for a universal mechanism underlyingthe hadronization In the next paragraph we recall theinterpretation of the string breaking as QCD H-U radiation

22 String Breaking and Event Horizon Let us outline thethermal hadron production process through H-U radiationfor the specific case of 119890+119890minus annihilation (see Figure 1) Theseparating primary 119902119902pair excites a further pair 119902

11199021from the

vacuum and this pair is in turn pulled apart by the primaryconstituents In the process the 119902

1shields the 119902 from its

original partner 119902 with a new 1199021199021string formed When it is

stretched to reach the pair production threshold a furtherpair is formed and so on [31 32] Such a pair productionmechanism is a special case ofH-U radiation [33ndash37] emitted

Advances in High Energy Physics 3

q

120574

q1 q1q2

q2q3

q

Figure 1 String breaking through 119902119902 pair production

as hadron 11990211199022when the quark 119902

1tunnels through its event

horizon to become 1199022 The corresponding hadron radiation

temperature is given by the Unruh form 119879119867= 1198862120587 where 119886

is the acceleration suffered by the quark 1199021due to the force

of the string attaching it to the primary quark 119876 This isequivalent to that suffered by quark 119902

2due to the effective

force of the primary antiquark 119876 Hence we have

119886119902=

120590

119908119902

=

120590

radic1198982

119902+ 1198962

119902

(2)

where 119908119902= radic119898

2

119902+ 1198962

119902is the effective mass of the produced

quark with 119898119902for the bare quark mass and 119896

119902the quark

momentum inside the hadronic system 11990211199021or 11990221199022 Since

the string breaks [25] when it reaches a separation distance

119909119902≃

2

120590

radic1198982

119902+ (

120587120590

2

) (3)

the uncertainty relation gives us 119896119902≃ 1119909

119902

119908119902= radic119898

2

119902+[

[

1205902

(41198982

119902+ 2120587120590)

]

]

(4)

for the effective mass of the quark The resulting quark-massdependent Unruh temperature is thus given by

119879 (119902119902) ≃

120590

2120587119908119902

(5)

Note that here it is assumed that the quark masses for 1199021and

1199022are equal For119898

119902≃ 0 (5) reduces to

119879 (00) ≃ radic

120590

2120587

(6)

as obtained in [25]If the produced hadron 119902

11199022consists of quarks of different

masses the resulting temperature has to be calculated asan average of the different accelerations involved For one

massless quark (119898119902≃ 0) and one of strange quark mass 119898

119904

the average acceleration becomes

1198860119904=

11990801198860+ 119908119904119886119904

1199080+ 119908119904

=

2120590

1199080+ 119908119904

(7)

From this the Unruh temperature of a strange meson is givenby

119879 (0119904) ≃

120590

120587 (1199080+ 119908119904)

(8)

with 1199080≃ radic12120587120590 and 119908

119904given by (4) with 119898

119902= 119898119904

Similarly we obtain

119879 (119904119904) ≃

120590

2120587119908119904

(9)

for the temperature of a meson consisting of a strange quark-antiquark pair (120601) With 120590 ≃ 02GeV2 (6) gives 119879

0≃

0178GeV A strange quark mass of 01 GeV reduces this to119879(0119904) ≃ 0167GeV and 119879(119904119904) ≃ 157MeV that is by about 6and 12 respectively

The scheme is readily generalized to baryonsTheproduc-tion pattern is illustrated in Figure 2 and leads to an averageof the accelerations of the quarks involved We thus have

119879 (000) = 119879 (0) ≃

120590

21205871199080

(10)

for nucleons

119879 (00119904) ≃

3120590

2120587 (21199080+ 119908119904)

(11)

for Λ and Σ production

119879 (0119904119904) ≃

3120590

2120587 (1199080+ 2119908119904)

(12)

for Ξ production and

119879 (119904119904119904) = 119879 (119904119904) ≃

120590

2120587119908119904

(13)

for that of Ωrsquos We thus obtain a resonance gas picture withfive different hadronization temperatures as specified by thestrangeness content of the hadron in question 119879(00) =

119879(000) 119879(0119904) 119879(119904119904) = 119879(119904119904119904) 119879(00119904) and 119879(0119904119904)

23 Strangeness Suppression in Elementary Collisions Inorder to evaluate the primary hadron multiplicities the pre-vious different species-dependent temperatures fully deter-mined by the string tension 120590 and the strange quark mass119898119904 have been inserted into a complete statistical model

calculation [28]Apart from possible variations of the quantities of 120590 and

119898119904 the description is thus parameter-free As illustration we

show in Table 1 the temperatures obtained for 120590 = 02GeV2and three different strange quark masses It is seen that inall cases the temperature for a hadron carrying nonzero


q

uu

u

d

q

d

d

uu

p

p

120574

Figure 2 Nucleon formation by string breaking through 119902119902 pairproduction

Table 1 Hadronization temperatures for hadrons of differentstrangeness content for 119898

119904= 0075 0100 0125GeV and 120590 =

02GeV2

119879 119898119904= 0075 119898

119904= 0100 119898

119904= 0125

119879(00) 0178 0178 0178119879(0119904) 0172 0167 0162119879(119904119904) 0166 0157 0148119879(000) 0178 0178 0178119879(00119904) 0174 0171 0167119879(0119904119904) 0170 0164 0157119879(119904119904119904) 0166 0157 0148

strangeness is lower than that of nonstrange hadrons andthis leads to an overall strangeness suppression in elementarycollisions in good agreement with data [28] without theintroduction of the ad-hoc parameter 120574

119904

Although a complete statistical hadronization modelcalculation has been necessary for a detailed comparisonwithexperimental results [28] a rather simple coarse grain argu-ment can illustrate why a reduction of the H-U temperaturefor strange particle production reproduces the 120574

119904effect

To simplify matters let us assume that there are only twospecies scalar and electrically neutral mesons ldquopionsrdquo withmass119898

120587 and ldquokaonsrdquo with mass119898

119896and strangeness 119904 = 1

According to the statistical model with the 120574119904suppression

factor the ratio119873119896119873120587is obtained by (1) and is given by

119873119896

119873120587

10038161003816100381610038161003816100381610038161003816

stat

120574119904

=

1198982

119896

1198982

120587

120574119904

1198702(119898119896119879)

1198702(119898120587119879)

(14)

because there is thermal equilibrium at temperature 119879On the other hand in the H-U based statistical model

there is no 120574119904 119879119896= 119879(0119904) = 119879

120587= 119879(00) = 119879 and therefore

119873119896

119873120587

10038161003816100381610038161003816100381610038161003816

stat

H-U=

1198982

119896

1198982

120587

119879119896

119879120587

1198702(119898119896119879119896)

1198702(119898120587119879120587)

(15)

Which corresponds to a 120574119904parameter given by

120574119904=

119879119896

119879120587

1198702(119898119896119879119896)

1198702(119898119896119879120587)

(16)

For 120590 = 02GeV2 119898119904= 01GeV 119879

120587= 178Mev and 119879

119896=

167Mev (see Table 1) the crude evaluation by (16) gives 120574119904≃

073In conclusion our picture implies that the produced

hadrons are emitted slightly ldquoout of equilibriumrdquo in the sensethat the emission temperatures are not identical As longas there is no final state interference between the producedquarks or hadrons we expect to observe this difference andhence amodification of the production of strange hadrons incomparison to the corresponding full equilibrium values

Once such interference becomes likely due to theincreases of parton density as in high energy heavy ioncollisions equilibrium can be essentially restored producinga small strangeness suppression as we will discuss in the nextsection

3 Hawking-Unruh Strangeness Enhancementin Heavy Ion Collisions

As previosuly discussed in elementary collisions the produc-tion of hadrons containing 119899 strange quarks or antiquarks isreduced in comparison to nucleus-nucleus scattering and thisreduction can be accounted for by the introduction of thephenomenological universal strangeness suppression factor120574119899

119904However the hadron production in high energy collisions

occurs in a number of causally disconnected regions offinite space-time size [38] As a result globally conservedquantumnumbers (charge strangeness and baryon number)must be conserved locally in spatially restricted correlationclusters This provides a dynamical basis for understandingthe suppression of strangeness production in elementaryinteractions (119901119901 119890+119890minus) due to a small strangeness correlationvolume [11 30 39 40]

The H-U counterpart of this mechanism is that inelementary collisions there is a small number of partonsin a causally connected region and the hadron productioncomes from the sequential breaking of independent 119902119902 stringswith the consequent species-dependent temperatures whichreproduce the strangeness suppression

In contrast the space-time superposition of many colli-sions in heavy ion interactions largely removes these causalityconstraints [38] resulting in an ideal hadronic resonance gasin full equilibrium

In the H-U approach the effect of a large number ofcausally connected quarks and antiquarks can be imple-mented by defining the average temperature of the systemanddetermining the hadronmultiplicities by the statistical modelwith this ldquoequilibriumrdquo temperature

Indeed the average temperature depends on the numbersof light quarks 119873

119897 and of strange quarks 119873

119904 which in

turn are counted by the number of strange and nonstrangehadrons in the final state at that temperature


A detailed analysis requires again a full calculation in thestatistical model that will be done in a forthcoming paperhowever the mechanism can be roughly illustrated in theworld of ldquopionsrdquo and ldquokaonsrdquo introduced in Section 2

Let us consider a high density system of quarks andantiquarks in a causally connected region Generalizing ourformulas in Section 2 the average acceleration is given by

119886 =

11987311989711990801198860+ 119873119904119908119904119886119904

1198731198971199080+ 119873119904119908119904

(17)

By assuming 119873119897≫ 119873119904 by ((6)ndash(8)) after a simple algebra

the average temperature 119879 = 1198862120587 turns out to be

119879 = 119879 (00) [1 minus

119873119904

119873119897

1199080+ 119908119904

1199080

(1 minus

119879 (0119904)

119879 (00)

)] + 119874[(

119873119904

119873119897

)

2

]

(18)

Now in our world of ldquopionsrdquo and ldquokaonsrdquo one has119873119897= 2119873120587+

119873119896and119873

119904= 119873119896and therefore

119879 = 119879 (00) [1 minus

119873119896

2119873120587

1199080+ 119908119904

1199080

(1 minus

119879 (0119904)

119879 (00)

)]

+ 119874[(

119873119896

119873120587

)

2

]

(19)

On the other hand in the H-U based statistical calculationthe ratio 119873

119896119873120587

depends on the equilibrium (average)temperature 119879 that is

119873119896

119873120587

=

1198982

119896

1198982

120587

1198702(119898119896119879)

1198702(119898120587119879)

(20)

and therefore one has to determine the temperature 119879in such a way that (19) and (20) are self-consistent Thiscondition implies the equation

2

[1 minus 119879119879 (00)]1199080

[1 minus 119879 (0119904) 119879 (00)] (119908119904+ 1199080)

=

1198982

119896

1198982

120587

1198702(119898119896119879)

1198702(119898120587119879)

(21)

that can be solved numericallyFor 120590 = 02GeV2 119898

119904= 01 and the temperatures in

Table 1 the average temperature turns out 119879 = 174Mev andone can evaluate the Wroblewski factor defined by

120582 =

2119873119904

119873119897

(22)

where119873119904is the number of strange and antistrange quarks in

the hadrons in the final state and 119873119897is the number of light

quarks and antiquarks in the final state minus their numberin the initial configuration

The experimental value of the Wroblewski factor in highenergy collisions is rather independent on the energy and isabout 120582 ≃ 026 in elementary collisions and 120582 ≃ 05 fornucleus-nucleus scattering

In our simplified world for 119890+119890minus annhilation 120582 = 119873119896119873120587

and is given by (15) with the species-dependent temperaturesin Table 1 One gets 120582 ≃ 026

To evaluate the Wroblewski factor in nucleus-nucleuscollisions one has to consider the average ldquoequilibriumrdquotemperature 119879 and the number of light quarks in the initialconfiguration The latter point requires a realistic calculationin the statitical model which includes all resonances andstable particles

However to show that one is on the right track let usneglect the problem of the initial configurationa and letus evaluate the effect of substituting in (15) the species-dependent temperatures with the equilibrium temperature119879 With this simple modification the Wroblewski factorincreases 120582 = 033

In other terms the change from a nonequilibrium condi-tion with species-dependent temperatures to an equilibratedsystem with the average temperature 119879 is able to reproducepart of the observed growing of the number of strange quarkswith respect to elementary interactions

4 Conclusions and Comments

The Hawking-Unruh hadronization mechanism gives adynamical basis to the universality of the temperatureobtained in the statistical model to the strangeness sup-pression in elementary collisions and to its enhancement innucleus-nucleus scatterings

We are aware that the simplified formulation in Section 3has to be checked by a full analysis in the hadron reso-nance gas framework or by montecarlo simulations and thatin addition the interpretation of production of charmedhadrons in 119890

+119890minus annihilations which is very successfully

described by the hadron resonance gas model with an exactcharm conservation should be addressed through Hawking-Unruh radiation mechanism to check if there is a consistentpicture

However the proposed approach is also able to under-stand the extremely fast equilibration of hadronic matterIndeed Hawking-Unruh radiation provides a stochasticrather than kinetic approach to equilibrium with a random-ization essentially due to the quantum physics of the colorhorizon The barrier to information transfer due to the eventhorizon requires that the resulting radiation states excitedfrom the vacuum be distributed according to maximumentropy with a temperature determined by the strength of theconfining field

The ensemble of all produced hadrons averaged overall events then leads to the same equilibrium distributionas obtained in hadronic matter by kinetic equilibration Inthe case of a very high energy collision with a high averagemultiplicity already one event can provide such equilibriumbecause of the interruption of information transfer at each ofthe successive quantum color horizons there is no phase rela-tion between two successive production steps in a given eventThe destruction of memory which in kinetic equilibrationis achieved through sufficiently many successive collisionsis here automatically provided by the tunnelling process So


the thermal hadronic final state in high energy collisions isobtained through a stochastic process

Finally high energy particle physics and in particularhadron production is in our opinion the promising sectorto find the analogue of the Hawking-Unruh radiation for twomain reasons color confinement and the huge accelerationthat cannot be reached in any other dynamical systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Becattini ldquoA thermodynamical approach to hadron produc-tion in 119890+119890minus collisionsrdquo Zeitschrift fur Physik C vol 69 pp 485ndash492 1996

[2] F Becattini ldquoUniversality of thermal hadron production in pp119901119901 and 119890+119890minus collisionsrdquo in Universality Features in MultihadronProduction and the Leading Effect pp 74ndash104 World ScientificSingapore 1998 httparxivorgabshep-ph9701275

[3] F Becattini and G Passaleva ldquoStatistical hadronization modeland transverse momentum spectra of hadrons in high energycollisionsrdquo European Physical Journal C vol 23 no 3 pp 551ndash583 2002

[4] F Becattini and U Heinz ldquoThermal hadron production in ppand 119901119901 collisionsrdquo Zeitschrift fur Physik C vol 76 no 268 p286 1997

[5] J Cleymans H Satz E Suhonen and D W Von OertzenldquoStrangeness production in heavy ion collisions at finite baryonnumber densityrdquo Physics Letters B vol 242 no 1 pp 111ndash1141990

[6] J Cleymans and H Satz ldquoThermal hadron production in highenergy heavy ion collisionsrdquo Zeitschrift fur Physik C vol 57 pp135ndash147 1993

[7] K Redlich J Cleymans H Satz and E Suhonen ldquoHadroni-sation of quark-gluon plasmardquo Nuclear Physics A vol 566 pp391ndash394 1994

[8] P Braun-Munzinger J Stachel J P Wessels and N Xu ldquoTher-mal equilibration and expansion in nucleus-nucleus collisionsat the AGSrdquo Physics Letters B vol 344 no 1ndash4 pp 43ndash48 1995

[9] F Becattini M Gazdzicki and J Sollfrank ldquoOn chemicalequilibrium in nuclear collisionsrdquo European Physical Journal Cvol 5 no 1 pp 143ndash153 1998

[10] F Becattini et al ldquoFeatures of particle multiplicities andstrangeness production in central heavy ion collisions between17A and 158A GeVcrdquo Physical Review C vol 64 Article ID024901 2001

[11] P Braun-Munzinger K Redlich and J Stachel ldquoParticle pro-duction in heavy ion collisionsrdquo inQuark-Gluon Plasma 3 R CHwa and X N Wang Eds World Scientific Singapore 2003

[12] F Becattini ldquoStatistical hadronisation phenomenologyrdquoNuclearPhysics A vol 702 pp 336ndash340 2001

[13] J Letessier J Rafelski and A Tounsi ldquoGluon productioncooling and entropy in nuclear collisionsrdquo Physical Review Cvol 50 no 1 pp 406ndash409 1994

[14] F Becattini J Manninen and M Gazdzicki ldquoEnergy andsystem size dependence of chemical freeze-out in relativisticnuclear collisionsrdquoPhysical ReviewC vol 73 Article ID 0449052006

[15] P Braun-Munzinger D Magestro K Redlich and J StachelldquoHadron production in Au-Au collisions at RHICrdquo PhysicsLetters B vol 518 no 1-2 pp 41ndash46 2001

[16] UHeinz ldquoPrimordial hadrosynthesis in the little bangrdquoNuclearPhysics A vol 661 pp 140ndash149 1999

[17] R Stock ldquoThe parton to hadron phase transition observed inPb + Pb collisions at 158GeV per nucleonrdquo Physics Letters Bvol 456 no 2-4 pp 277ndash282 1999

[18] A Bialas ldquoFluctuations of the string tension and transversemass distributionrdquo Physics Letters B vol 466 no 2-4 pp 301ndash304 1999

[19] H Satz ldquoThe search for the QGP a critical appraisalrdquo NuclearPhysics BmdashProceedings Supplements vol 94 no 1ndash3 pp 204ndash218 2001

[20] J Hormuzdiar S DHHsu andGMahlon ldquoParticlemultiplic-ities and thermalization in high energy collisionsrdquo InternationalJournal of Modern Physics E vol 12 no 5 pp 649ndash659 2003

[21] V Koch ldquoSome remarks on the statistical model of heavy ioncollisionsrdquo Nuclear Physics A vol 715 p 108 2003

[22] L McLerran ldquoThe quark gluon plasma and the color glasscondensate 4 lecturesrdquo httparxivorgabshep-ph0311028

[23] Y Dokshitzer ldquoHistorical and futuristic perturbative and non-perturbative aspects of QCD jet physicsrdquo Acta Physica PolonicaB vol 36 p 361 2005

[24] F Becattini ldquoWhat is the meaning of the statistical hadroniza-tion modelrdquo Journal of Physics Conference Series vol 5 p 1752005

[25] P Castorina D Kharzeev and H Satz ldquoThermal hadronizationand Hawking-Unruh radiation in QCDrdquo European PhysicalJournal C vol 52 no 1 pp 187ndash201 2007

[26] S W Hawking ldquoParticle creation by black holesrdquo Communica-tions in Mathematical Physics vol 43 no 3 pp 199ndash220 1975

[27] W G Unruh ldquoNotes on black-hole evaporationrdquo PhysicalReview D vol 14 no 4 pp 870ndash892 1976

[28] F Becattini P Castorina J Manninen and H Satz ldquoThethermal production of strange and non-strange hadrons in 119890+119890minusCollisionsrdquo European Physical Journal C vol 56 no 4 pp 493ndash510 2008

[29] A Wroblewski ldquoOn the strange quark suppression factor inhigh energy collisionsrdquo Acta Physica Polonica B vol 16 p 3791985

[30] RHagedorn andKRedlich ldquoStatistical thermodynamics in rel-ativistic particle and ion physics canonical or grand canonicalrdquoZeitschrift fur Physik C vol 27 pp 541ndash551 1985

[31] J D Bjorken ldquoHadron final states in deep inelastic processesrdquoinCurrent Induced Reactions vol 56 of Lecture Notes in Physicspp 93ndash158 Springer 1976

[32] A Casher H Neuberger and S Nussinov ldquoChromoelectric-flux-tube model of particle productionrdquo Physical Review D vol20 no 1 pp 179ndash188 1979

[33] R Brout R Parentani and P Spindel ldquoThermal propertiesof pairs produced by an electric field a tunneling approachrdquoNuclear Physics B vol 353 no 1 pp 209ndash236 1991

[34] R Parentani and S Massar ldquoSchwinger mechanism Unruheffect and production of accelerated black holesrdquo PhysicalReview D vol 55 no 6 pp 3603ndash3613 1997

[35] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 1999


[36] D Kharzeev and K Tuchin ldquoFrom color glass condensate toquark-gluonplasma through the event horizonrdquoNuclear PhysicsA vol 753 no 3-4 pp 316ndash334 2005

[37] S P Kim ldquoSchwinger mechanism and Hawking radiation asquantum tunnelingrdquo Journal of the Korean Physical Society vol53 pp 1095ndash1099 2008

[38] P Castorina and H Satz ldquoCausality constraints on hadronproduction in high energy collisionsrdquo International Journal ofModern Physics E vol 23 Article ID 1450019 2014

[39] I Kraus J Cleymans H Oeschler K Redlich and S WheatonldquoChemical equilibrium in collisions of small systemsrdquo PhysicalReview C vol 76 Article ID 064903 2007

[40] I Kraus J Cleymans H Oeschler and K Redlich ldquoParticleproduction in p-p collisions and predictions for radic119904 = 14TeVat the CERN Large Hadron Collider (LHC) rdquo Physical ReviewC vol 79 Article ID 014901 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


due to colour confinement In the case of approximatelymassless quarks the resulting hadronization temperature isdetermined by the string tension 120590 with 119879 ≃ radic1205902120587 ≃

170Mev [25] Moreover in [28] it has been shown thatstrangeness suppression in elementary collisions naturallyoccurs in this framework without requiring a specific sup-pression factorThe crucial role here is played by the nonneg-ligible strange quark mass which modifies the emission tem-perature for such quarks in the Hawking-Unruh approachthe temperature associated with strange particle productionis different from that one of nonstrange particles Althoughthis conclusion has been obtained by a full analysis in thestatistical hadronization model in [28] here we propose anelementary scheme of understanding strangeness suppres-sionwhichmoreover opens the possibility of explaining whythis suppression disappears in heavy ion collisions where thelarge parton density in causally connected space-time regionof hadronization produces a unique temperature for strangeand nonstrange particles

In particular in the second section after recalling thestatistical hadronizationmodel one discusses the subsequentdescription in terms of H-U radiation in QCD includingthe dependence of the radiation temperature on the massof the produced quark and the strangeness suppression inelementary collisions In Section 3 the dynamical mechanismfor strangeness enhancement (ie no suppression) in rela-tivistic heavy ion scatterings is introduced giving a coarsegrain evaluation of the Wroblewski factor [29] a parameterwhich counts the relative number of strange and nonstrangequarks and antiquarks Section 4 is devoted to comments andconclusions

2 Hawking-Unruh Hadronization

In this section we will recall the essentials of the statisticalhadronization model and of the Hawking-Unruh analysis ofthe string breaking mechanism For a detailed descriptionssee [3 25 28]

21 Statistical Hadronization Model The statistical hadr-onization model assumes that hadronization in high energycollisions is a universal process proceeding through theformation ofmultiple colourlessmassive clusters (or fireballs)of finite spacial extension These clusters are taken to decayinto hadrons according to a purely statistical law everymultihadron state of the cluster phase space defined by itsmass volume and charges is equally probable The massdistribution and the distribution of charges (electric bary-onic and strange) among the clusters and their (fluctuating)number are determined in the prior dynamical stage of theprocess Once these distributions are known each cluster canbe hadronized on the basis of statistical equilibrium leadingto the calculation of averages in themicrocanonical ensembleenforcing the exact conservation of energy and charges ofeach cluster

Hence in principle one would need the mentioneddynamical information in order to make definite quantitativepredictions to be compared with data Nevertheless for

Lorentz-invariant quantities such as multiplicities one canintroduce a simplifying assumption and thereby obtain asimple analytical expression in terms of a temperature Thekey point is to assume that the distribution of masses andcharges among clusters is again purely statistical [3] so thatas far as the calculation of multiplicities is concerned theset of clusters becomes equivalent on average to a largecluster (equivalent global cluster) whose volume is the sumof proper cluster volumes and whose charge is the sum ofcluster charges (and thus the conserved charge of the initialcolliding system) In such a global averaging process theequivalent cluster generally turns out to be large enough inmass and volume so that the canonical ensemble becomesa good approximation In other words a temperature canbe introduced which replaces the a priorimore fundamentaldescription in terms of an energy density

To obtain a simple expression for our further discussionwe neglect for the moment an aspect which is important inany actual analysis Although in elementary collisions theconservation of the various discrete abelian charges (electriccharge baryon number strangeness heavy flavour) has tobe taken into account exactly [30] we here consider forthe moment a grand-canonical picture We also assumeBoltzmann distributions for all hadrons The multiplicity ofa given scalar hadronic species 119895 then becomes

⟨119899119895⟩primary

=

1198811198791198982

119895

21205872120574

119899119895

119904 1198702(

119898119895

119879

)(1)

with 119898119895denoting its mass and 119899

119895the number of strange

quarksantiquarks it contains Here primary indicates thatit gives the number at the hadronisation point prior to allsubsequent resonance decay The Hankel function 119870

2(119909)

with 119870(119909) sim expminus119909 for large 119909 gives the Boltzmannfactor while119881 denotes the overall equivalent cluster volumeIn other words in an analysis of 4120587 data of elementarycollisions 119881 is the sum of the all cluster volumes at alldifferent rapidities It thus scales with the overall multiplicityand hence increases with collision energy A fit of produc-tion data based on the statistical hadronisation model thusinvolves three parameters the hadronisation temperature 119879the strangeness suppression factor 120574

119904 and the equivalent

global cluster volume 119881As previously discussed at high energy the temperature

turns out to be independent of the initial configurationand this result calls for a universal mechanism underlyingthe hadronization In the next paragraph we recall theinterpretation of the string breaking as QCD H-U radiation

22 String Breaking and Event Horizon Let us outline thethermal hadron production process through H-U radiationfor the specific case of 119890+119890minus annihilation (see Figure 1) Theseparating primary 119902119902pair excites a further pair 119902

11199021from the

vacuum and this pair is in turn pulled apart by the primaryconstituents In the process the 119902

1shields the 119902 from its

original partner 119902 with a new 1199021199021string formed When it is

stretched to reach the pair production threshold a furtherpair is formed and so on [31 32] Such a pair productionmechanism is a special case ofH-U radiation [33ndash37] emitted


q

120574

q1 q1q2

q2q3

q










119886119902=

120590

119908119902

=

120590

radic1198982

119902+ 1198962

119902

(2)

where 119908119902= radic119898

2

119902+ 1198962



119902the quark



119909119902≃

2

120590

radic1198982

119902+ (

120587120590

2

) (3)


119902

119908119902= radic119898

2

119902+[

[

1205902

(41198982

119902+ 2120587120590)

]

]

(4)


119879 (119902119902) ≃

120590

2120587119908119902

(5)



119902≃ 0 (5) reduces to

119879 (00) ≃ radic

120590

2120587

(6)





119904


1198860119904=

11990801198860+ 119908119904119886119904

1199080+ 119908119904

=

2120590

1199080+ 119908119904

(7)


119879 (0119904) ≃

120590

120587 (1199080+ 119908119904)

(8)

with 1199080≃ radic12120587120590 and 119908

119904given by (4) with 119898

119902= 119898119904

Similarly we obtain

119879 (119904119904) ≃

120590

2120587119908119904

(9)


0≃



119879 (000) = 119879 (0) ≃

120590

21205871199080

(10)

for nucleons

119879 (00119904) ≃

3120590

2120587 (21199080+ 119908119904)

(11)


119879 (0119904119904) ≃

3120590

2120587 (1199080+ 2119908119904)

(12)


119879 (119904119904119904) = 119879 (119904119904) ≃

120590

2120587119908119904

(13)


119879(000) 119879(0119904) 119879(119904119904) = 119879(119904119904119904) 119879(00119904) and 119879(0119904119904)






q

uu

u

d

q

d

d

uu

p

p

120574



119904= 0075 0100 0125GeV and 120590 =

02GeV2

119879 119898119904= 0075 119898

119904= 0100 119898

119904= 0125

119879(00) 0178 0178 0178119879(0119904) 0172 0167 0162119879(119904119904) 0166 0157 0148119879(000) 0178 0178 0178119879(00119904) 0174 0171 0167119879(0119904119904) 0170 0164 0157119879(119904119904119904) 0166 0157 0148


119904


119904effect






119873119896

119873120587

10038161003816100381610038161003816100381610038161003816

stat

120574119904

=

1198982

119896

1198982

120587

120574119904

1198702(119898119896119879)

1198702(119898120587119879)

(14)


there is no 120574119904 119879119896= 119879(0119904) = 119879

120587= 119879(00) = 119879 and therefore

119873119896

119873120587

10038161003816100381610038161003816100381610038161003816

stat

H-U=

1198982

119896

1198982

120587

119879119896

119879120587

1198702(119898119896119879119896)

1198702(119898120587119879120587)

(15)


120574119904=

119879119896

119879120587

1198702(119898119896119879119896)

1198702(119898119896119879120587)

(16)

For 120590 = 02GeV2 119898119904= 01GeV 119879

120587= 178Mev and 119879

119896=














119904 which in





119886 =

11987311989711990801198860+ 119873119904119908119904119886119904

1198731198971199080+ 119873119904119908119904

(17)



119879 = 119879 (00) [1 minus

119873119904

119873119897

1199080+ 119908119904

1199080

(1 minus

119879 (0119904)

119879 (00)

)] + 119874[(

119873119904

119873119897

)

2

]

(18)


119873119896and119873

119904= 119873119896and therefore

119879 = 119879 (00) [1 minus

119873119896

2119873120587

1199080+ 119908119904

1199080

(1 minus

119879 (0119904)

119879 (00)

)]

+ 119874[(

119873119896

119873120587

)

2

]

(19)


119896119873120587


119873119896

119873120587

=

1198982

119896

1198982

120587

1198702(119898119896119879)

1198702(119898120587119879)

(20)


2

[1 minus 119879119879 (00)]1199080

[1 minus 119879 (0119904) 119879 (00)] (119908119904+ 1199080)

=

1198982

119896

1198982

120587

1198702(119898119896119879)

1198702(119898120587119879)

(21)




120582 =

2119873119904

119873119897

(22)






















References















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




q

120574

q1 q1q2

q2q3

q










119886119902=

120590

119908119902

=

120590

radic1198982

119902+ 1198962

119902

(2)

where 119908119902= radic119898

2

119902+ 1198962



119902the quark



119909119902≃

2

120590

radic1198982

119902+ (

120587120590

2

) (3)


119902

119908119902= radic119898

2

119902+[

[

1205902

(41198982

119902+ 2120587120590)

]

]

(4)


119879 (119902119902) ≃

120590

2120587119908119902

(5)



119902≃ 0 (5) reduces to

119879 (00) ≃ radic

120590

2120587

(6)





119904


1198860119904=

11990801198860+ 119908119904119886119904

1199080+ 119908119904

=

2120590

1199080+ 119908119904

(7)


119879 (0119904) ≃

120590

120587 (1199080+ 119908119904)

(8)

with 1199080≃ radic12120587120590 and 119908

119904given by (4) with 119898

119902= 119898119904

Similarly we obtain

119879 (119904119904) ≃

120590

2120587119908119904

(9)


0≃



119879 (000) = 119879 (0) ≃

120590

21205871199080

(10)

for nucleons

119879 (00119904) ≃

3120590

2120587 (21199080+ 119908119904)

(11)


119879 (0119904119904) ≃

3120590

2120587 (1199080+ 2119908119904)

(12)


119879 (119904119904119904) = 119879 (119904119904) ≃

120590

2120587119908119904

(13)


119879(000) 119879(0119904) 119879(119904119904) = 119879(119904119904119904) 119879(00119904) and 119879(0119904119904)






q

uu

u

d

q

d

d

uu

p

p

120574



119904= 0075 0100 0125GeV and 120590 =

02GeV2

119879 119898119904= 0075 119898

119904= 0100 119898

119904= 0125

119879(00) 0178 0178 0178119879(0119904) 0172 0167 0162119879(119904119904) 0166 0157 0148119879(000) 0178 0178 0178119879(00119904) 0174 0171 0167119879(0119904119904) 0170 0164 0157119879(119904119904119904) 0166 0157 0148


119904


119904effect






119873119896

119873120587

10038161003816100381610038161003816100381610038161003816

stat

120574119904

=

1198982

119896

1198982

120587

120574119904

1198702(119898119896119879)

1198702(119898120587119879)

(14)


there is no 120574119904 119879119896= 119879(0119904) = 119879

120587= 119879(00) = 119879 and therefore

119873119896

119873120587

10038161003816100381610038161003816100381610038161003816

stat

H-U=

1198982

119896

1198982

120587

119879119896

119879120587

1198702(119898119896119879119896)

1198702(119898120587119879120587)

(15)


120574119904=

119879119896

119879120587

1198702(119898119896119879119896)

1198702(119898119896119879120587)

(16)

For 120590 = 02GeV2 119898119904= 01GeV 119879

120587= 178Mev and 119879

119896=














119904 which in





119886 =

11987311989711990801198860+ 119873119904119908119904119886119904

1198731198971199080+ 119873119904119908119904

(17)



119879 = 119879 (00) [1 minus

119873119904

119873119897

1199080+ 119908119904

1199080

(1 minus

119879 (0119904)

119879 (00)

)] + 119874[(

119873119904

119873119897

)

2

]

(18)


119873119896and119873

119904= 119873119896and therefore

119879 = 119879 (00) [1 minus

119873119896

2119873120587

1199080+ 119908119904

1199080

(1 minus

119879 (0119904)

119879 (00)

)]

+ 119874[(

119873119896

119873120587

)

2

]

(19)


119896119873120587


119873119896

119873120587

=

1198982

119896

1198982

120587

1198702(119898119896119879)

1198702(119898120587119879)

(20)


2

[1 minus 119879119879 (00)]1199080

[1 minus 119879 (0119904) 119879 (00)] (119908119904+ 1199080)

=

1198982

119896

1198982

120587

1198702(119898119896119879)

1198702(119898120587119879)

(21)




120582 =

2119873119904

119873119897

(22)






















References















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




q

uu

u

d

q

d

d

uu

p

p

120574



119904= 0075 0100 0125GeV and 120590 =

02GeV2

119879 119898119904= 0075 119898

119904= 0100 119898

119904= 0125

119879(00) 0178 0178 0178119879(0119904) 0172 0167 0162119879(119904119904) 0166 0157 0148119879(000) 0178 0178 0178119879(00119904) 0174 0171 0167119879(0119904119904) 0170 0164 0157119879(119904119904119904) 0166 0157 0148


119904


119904effect






119873119896

119873120587

10038161003816100381610038161003816100381610038161003816

stat

120574119904

=

1198982

119896

1198982

120587

120574119904

1198702(119898119896119879)

1198702(119898120587119879)

(14)


there is no 120574119904 119879119896= 119879(0119904) = 119879

120587= 119879(00) = 119879 and therefore

119873119896

119873120587

10038161003816100381610038161003816100381610038161003816

stat

H-U=

1198982

119896

1198982

120587

119879119896

119879120587

1198702(119898119896119879119896)

1198702(119898120587119879120587)

(15)


120574119904=

119879119896

119879120587

1198702(119898119896119879119896)

1198702(119898119896119879120587)

(16)

For 120590 = 02GeV2 119898119904= 01GeV 119879

120587= 178Mev and 119879

119896=














119904 which in





119886 =

11987311989711990801198860+ 119873119904119908119904119886119904

1198731198971199080+ 119873119904119908119904

(17)



119879 = 119879 (00) [1 minus

119873119904

119873119897

1199080+ 119908119904

1199080

(1 minus

119879 (0119904)

119879 (00)

)] + 119874[(

119873119904

119873119897

)

2

]

(18)


119873119896and119873

119904= 119873119896and therefore

119879 = 119879 (00) [1 minus

119873119896

2119873120587

1199080+ 119908119904

1199080

(1 minus

119879 (0119904)

119879 (00)

)]

+ 119874[(

119873119896

119873120587

)

2

]

(19)


119896119873120587


119873119896

119873120587

=

1198982

119896

1198982

120587

1198702(119898119896119879)

1198702(119898120587119879)

(20)


2

[1 minus 119879119879 (00)]1199080

[1 minus 119879 (0119904) 119879 (00)] (119908119904+ 1199080)

=

1198982

119896

1198982

120587

1198702(119898119896119879)

1198702(119898120587119879)

(21)




120582 =

2119873119904

119873119897

(22)






















References















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119886 =

11987311989711990801198860+ 119873119904119908119904119886119904

1198731198971199080+ 119873119904119908119904

(17)



119879 = 119879 (00) [1 minus

119873119904

119873119897

1199080+ 119908119904

1199080

(1 minus

119879 (0119904)

119879 (00)

)] + 119874[(

119873119904

119873119897

)

2

]

(18)


119873119896and119873

119904= 119873119896and therefore

119879 = 119879 (00) [1 minus

119873119896

2119873120587

1199080+ 119908119904

1199080

(1 minus

119879 (0119904)

119879 (00)

)]

+ 119874[(

119873119896

119873120587

)

2

]

(19)


119896119873120587


119873119896

119873120587

=

1198982

119896

1198982

120587

1198702(119898119896119879)

1198702(119898120587119879)

(20)


2

[1 minus 119879119879 (00)]1199080

[1 minus 119879 (0119904) 119879 (00)] (119908119904+ 1199080)

=

1198982

119896

1198982

120587

1198702(119898119896119879)

1198702(119898120587119879)

(21)




120582 =

2119873119904

119873119897

(22)






















References















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








References















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article hawking-unruh hadronization and ...research article hawking-unruh hadronization and...

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