ReviewArticle Balance Function in High-Energy Collisions · are now in order. The rapidity...

Review ArticleBalance Function in High-Energy Collisions

A Tawfik12 and Asmaa G Shalaby23

1Egyptian Center for Theoretical Physics (ECTP) Modern University for Technology and Information (MTI) Cairo 11571 Egypt2World Laboratory for Cosmology and Particle Physics (WLCAPP) Cairo Egypt3Physics Department Faculty of Science Benha University Benha 13815 Egypt

Correspondence should be addressed to A Tawfik atawfikengmtiedueg

Received 2 January 2015 Revised 13 March 2015 Accepted 16 March 2015

Academic Editor Ming Liu

Copyright copy 2015 A Tawfik and A G Shalaby 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

Aspects and implications of the balance functions (BF) in high-energy physics are reviewed The various calculations andmeasurements depending on different quantities for example system size collisions centrality and beam energy are discussedFirst the different definitions including advantages and even short-comings are highlighted It is found that BF which are mainlypresented in terms of relative rapidity and relative azimuthal and invariant relative momentum are sensitive to the interactioncentrality but not to the beam energy and can be used in estimating the hadronization time and the hadron-quark phase transitionFurthermore the quark chemistry can be determinedThe chemical evolution of the new-state-of-matter the quark-gluon plasmaand its temporal-spatial evolution femtoscopy of two-particle correlations are accessibleThe production time of positive-negativepair of charges can be determined from the widths of BF Due to the reduction in the diffusion time narrowed widths refer todelayed hadronization It is concluded that BF are powerful tools characterizing hadron-quark phase transition and estimatingsome essential properties

1 Introduction

The quark-gluon plasma (QGP) a state of matter created at01ndash1 120583s after the Big Bang is believed to be discovered in therelativistic heavy-ion collider (RHIC) at BNL ten years ago[1ndash5] The heavy-ion program at the large hadron collider(LHC) at CERN was designed to explore among othersthe properties of QGP In such sophisticated experimentalfacilities the nucleus-nucleus collisions at ultrarelativisticenergies are devoted to characterize the dynamical processesby whichmatter at extreme temperatures is produced and thefundamental properties that thismatter exhibits Over the lastfour decades various high-energy experiments using nucleonand nucleus beams have been evaluated Based on Bjorkenmodel the latter are likely able to produce a new-state-of-matter with partonic degrees of freedom where quarksand gluons deconfine forming a state similar to the plasmastate in atomic physics thus called quark-gluon plasma(QGP) In early Universe QGP is believed to entirely fillthe cosmological background geometry Furthermore theextreme conditions available inside the cores of compact

stars are likely able to compress the hadronic matter Suchextreme compression has the same effect as that of extremetemperature Both are necessary to derive the confinedhadrons into deconfined partons The temporal and spatialevolution of hotmatter till the creation of hadrons is sketchedin Figure 1

The discovery ofQGP imposes extreme experimental andtheoretical challenges and is a good example about physicalproblems which should wait even decades for their properexplanation [6] One of the main QGP signatures is the 119869120595suppression that was proposed in 1986 byMatsui and Satz [7]During three decades the theoretical interpretation is stillunder debate [8ndash10] Other challenges can be summarized asfollows

(i) Mechanism of the elliptic flow there is an uncon-firmed point-of-view about the scaling of constituentquarks which is still not perfect because the resultsare not dealing directly with the constituents quarks[11]

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 186812 24 pageshttpdxdoiorg1011552015186812

2 Advances in High Energy Physics

t

120591f

Hadron gas

QGP

BeamBeam

1205910

z

QCD phasetransition

ThermalizationParton cascade

Kinetic freeze-out

Chemical freeze-out

Figure 1The space-time evolution of heavy-ion collisionThefigureis taken from [28]

(ii) Lattice QCD results predicted two orders for phasetransition(s) It is argued that a first-order phasetransition is likely in system consisting of two flavorswhile a second-order one is likely in the three-flavorsystem Furthermore a smooth cross-over was seenin the QCD simulations Linking such theoreticalpredictions with the experimental results would bepossible through varying the critical temperature Forinstance at low temperature the matter is confinedthat is hadronic phase while at high temperatureQGP phase is likely [12]

(iii) The strangeness enhancement at alternating gradientsynchrotron (AGS) is found larger than that at superproton synchrotron (SPS) which obviously seems toweaken the concept of strangeness enhancement asa signal of QGP [13] Nevertheless the search forenhancement at RHIC and LHC energies should becontinued

(iv) The estimation of the time span till equilibrationrefers to very small value (sim10 fmc) Thus the evo-lution of the equilibrated states cannot be evident[14] Thus it would not be possible to assure thatthe hadronic phase was originated in a partonic state(prior to hadronization) [14] The situation becomesmore drastic at RHIC and LHC energies The criticaland freeze-out temperatures become almost indistin-guishable [14]

The balance functions (BF) were proposed by Bass et al[15] as a measure for the correlation of the positive and nega-tive charged particles produced during the relativistic heavy-ion collisionsTheir width can be related to the hadronizationtime The charge correlation functions which are devoted tostudy the jets hadronization [16] are used to derive BF Sofar they have been estimated in pp collisions at intersectingstorage rings (ISR) [17ndash19] e+ + eminus annihilation at PETRA atDESY [20ndash24] Au+Au in STAR experiment at BNL RHIC[25] and Pb+Pb in NA49 experiment at CERN SPS [26 27]Due to charge conservation oppositely charged particles areproduced in pairs But the produced pairs are separated in therapidity region due to their different momenta This implies

that BF can be extracted from the fact that the pairs ofopposite charges are created in the local space This ideadefines how to proceed with the measurement of balancebetween produced pairs

The different heavy-ion experiments can be differentiatedaccording to the collision energy or nucleon-nucleon (NN)center-of-mass energyradic119904NN [46] the system size and type ofreactants whether being elementary NN or nucleus-nucleus(AA) collisions

119910 =1

2ln(

119864 + 119901119871

119864 minus 119901119871

) = ln(119864 + 119901

119871

119898perp

) (1)

where119901119871is the longitudinalmomentumand119898

perp= radic1198982 + 1199012

119879

is the transverse mass The Lorentz boosts are the trans-formations with respect to one of three dimensions takingas the frame of reference At ultrarelativistic energies itis convenient to deal with the pseudorapidity 120578 which isdefined in analogy to 119910 (1)

120578 = minus ln [tan(1205792)] (2)

where 120579 is the angle of emitted particles relative to the beamaxis

The present work is organized as follows Section 1presents a general overview about the history of QGPSection 2 is devoted to the various definitions of BF Theexperimental measurements will be discussed in Section 3Section 4 discusses some effective models used to calculateBF in high-energy physics Finally Section 5 presents thediscussion and conclusions

2 Definitions

In relativistic heavy-ion collisions it is assumed that manyproduced particles of different charges expand in temporaland spatial dimensions [39] Due to charge conservationboth positive and negative charges have to be produced inthe same space-time during the evolution of the mediumThe correlation between the opposite charges is characterizedthrough BF which apparently measure the balance betweenboth types of charges [47] In early studies Bass et al [15] haveproposed that BF are signatures differentiating between early-and late-stage of the hadronization The balance functionsare proposed to work as a ldquoclockrdquo determining whether thequark production occurred at early times 119905 lt 1 fmc orat late-stage [15] For charges created in the early stagebalancing charges are separated by the order of one unit ofrapidity while those formed in a late stage are far from thecorrelation Delayed hadronizationmeans that theQGP staysfor a long time This implies that the QGP might be formedat a certain time before the evolution of the hot matter Inprincipal BF were proposed to investigate the hadronizationfrom jets production in proton-proton collisions [17 18] Ina series of papers [17 18 48] BF were associated with chargecorrelations

Furthermore the conditional probability is the probabil-ity that an event will occur under some conditions while

Advances in High Energy Physics 3

another event is predicted to occur or to have occurred[49] According to the conditional probability a particle withcharge 119886 produced within a rapidity interval 119910

119886should be

accompanied by another particle with charge 119887 separatedfrom 119886 by a specified rapidity difference Δ119910 or 120575119910 = 119910

119887minus 119910

119886

The balance functions are defined as the linear combinationof these conditional probabilities [49] In terms of differentquantities such as azimuthal angle 120601 rapidity difference Δ119910pseudorapidity difference Δ120578 and invariant momentum 119902invBF can be expressed (3) [28 30]

(i) The balance functions are defined as [15]

119861 (1198751| 119875

2) equiv1

2120588 (119887 119875

2| 119886 119875

1) minus 120588 (119887 119875

2| 119887 119875

1)

+ 120588 (119886 1198752| 119887 119875

1) minus 120588 (119886 119875

2| 119886 119875

1)

(3)

where 120588(119887 1198752| 119886 119875

1) is the conditional probability

of finding particle of type 119887 in a bin at momentum1198752accompanied with another particle in a bin 119886

with momentum 1198751 119886 and 119887 are two typesvariables

like positive and negative charges For all chargedhadrons BF should be normalized in order to high-light the charge conservation conditionIn terms of rapidity distributions the balance func-tions can be defined as [39]

119861 (1198751| 119875

2) equiv1

2

119873minus+(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

+minus(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

2)

(4)

where 119873minus+(119875

1 119875

2) denotes the number of charged

particle pair (momenta of the observed positive andnegative charges) In a similar way the numberof positive (negative) pair charges for the differentdistributions reads119873

+minus119873

minusminusand119873

++

In an equivalent expression BF can be given as [50]

119861 (Δ2| Δ

1) =119863 (+ Δ

2| minus Δ

1) minus 119863 (+ Δ

2| + Δ

1)

119873+(Δ

1)

+119863 (minus Δ

2| + Δ

1) minus 119863 (minus Δ

2| minus Δ

1)

119873minus(Δ

1)

(5)

where 119873 and 119863 refer to the single and double (pair)particle functions In literature the distribution ofdouble and single particle is expressed in differentforms 119863(+ Δ

2| minus Δ

1)119873

+(Δ

1) or 120588(119887 Δ

2| 119886 Δ

1) =

119873(119887 Δ2| 119886 Δ

1)119873(119886 Δ

1) in which 119886 and 119887 are the

positive and negative charges [30]

(a) Rapidity dependence [50]

119873119894(Δ) = int

Δ

119889119899119894

119889119910119889119910

119863 (119894 Δ2| 119895 Δ

1) = int

Δ2

1198891199102intΔ1

1198891199101

1198892119899119894119895

1198891199101119889119910

2

(6)

where 119889119899119889119910 is the particle density Someremarks on the STAR measurements forinstance for the charge balance functionsare now in order The rapidity acceptanceranges between 119910

1ge minusΔ and 119910

2le Δ and

the pseudorapidity differences are kept con-stant while the pairs of produced particlesare detected In this regard notations like1199101minus 119910

2equiv Δ (119910

1+ 119910

2)2 equiv 119911 were introduced

[50](b) Momentum dependence [30]

120588 (119887 Δ2| 119886 Δ

1) =119873 (119887 Δ

2| 119886 Δ

1)

119873 (119886 Δ1)

(7)

where

119873(119887 Δ2| 119886 Δ

1) = int

Δ1

1198893

1199011intΔ2

1198893

1199012119891119886119887(119901

1 119901

2)

119873 (119886 Δ1) = int

Δ1

1198893

1199011119891119886(119901

1)

(8)

where 119891119886(119901

1) or 119891

119887(119901

1) are the single particle

distribution function and 119891119886(119901

1 119901

2) is the two-

particle (joint) momentum distribution

The joint momentum distributions 119891119886119887(119901

1 119901

2) can

be classified into quark-antiquark quark-quark orantiquarks created pairs These distributions are theproduct of the corresponding single particle momentumdistribution [30]

11989111990211199022

(1199011 119901

2) = 119873

1199021

1198731199022

1198991199021

(1199011) 119899

1199022

(1199012) (1 minus 120575

11990211199022

)

+ 1198731199021

(1198731199022

minus 1) 1198991199022

(1199012) 120575

11990211199022

11989111990211199022

(1199011 119901

2) = 119873

1199021

1198731199022

1198991199021

(1199011) 119899

1199022

(1199012) (1 minus 120575

11990211199022

)

+ 1198731199021

(1198731199022

minus 1) 1198991199022

(1199012) 120575

11990211199022

(9)

where 1199021and 119902

2are the quarks flavorsThe subscripts

119886 and 119887 refer to the quark-pair antiquark-pair orquark-antiquark pair The distribution of the quark-antiquark is given as

11989111990211199022

(1199011 119901

2)

= 1198731199021

1198731199022

1198991199021

(1199011) 119899

1199022

(1199012)

+ 1198731199021

[119899pair119902119902(119901

1 119901

2) minus 119899

1199021

(1199011) 119899

1199022

(1199012)] 120575

11990211199022

(10)

The single particle distribution for bosons and ferm-ions reads [46]

119891119865119861(119864 120573 120582 120574) =

1

120574minus1120582minus1119890120573119864 plusmn 1 (11)


where the dispersion relation reads 119864 = radic1199012 + 1198982120573 = 1119879 the fugacity 120582 = 119890120583119879 and 120574 is a Lagrangemultiplier related to the conservation of the numberof members of the ensemble In the same matterthe single particle distribution for antiquarks can beexpressed in terms of 119899pair

119902119902

119899119902(119901

2) = int119889

3

1199011119899pair119902119902(119901

1 119901

2) (12)

With this regard the following frames should bedefined

(1) laboratory frame is the inertial reference framewith the coordinates 119905 119909 119910 and 119911

(2) comoving frame at a time 1199050 this is the iner-

tial frame in which the accelerated observer isinstantaneously at rest at 119905 = 119905

0 Thus the term

ldquocomoving framerdquo refers to a different frame ateach 119905

0

It is argued that the physical quantities which are sig-nificant and meaningful are the ones correspondingto the laboratory frameThismeans that the quantitiesare conserved only with respect to laboratory framebecause the comoving frame is an accelerated refer-ence frame [51] In comoving frame the single particlemomentum distribution for quarks or antiquarks inBoltzmann limit is given as [30]

119899lowast

119902(119901

lowast

) = 119899th (119901lowast

) =119890minus119864lowast119879

412058711989821198791198702(119898119879)

(13)

where astride refers to the quantities in the comovingframe

(ii) Uniform binning for charge a the multiplicity canbe determined from 119899

119886119894(120575119910) where 119910 is the rapidity

axis of the bin with the acceptance Δ119910 The bin sizeis 120575119910 and the bin number is 119872(Δ119910 120575119910) The totalmultiplicity reads [49]

119872(Δ119910120575119910)

sum

119894=1

119899119886119894(120575119910) equiv 119873

119886(Δ119910) (14)

The bin counts represent integrals of the form

119899119886119894(120575119910) asymp int

119910119894+1205751199102

119910119894minus1205751199102

1205881119899119886(119910) 119889119910 (15)

where 119899119886(119910) is the number density of a single-particle

distribution determined from the histogram of

the ensemble averages and 119899119886119894(120575119910) Thus BF are

defined as [49]

119861119870(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus 1198861198871

sum119872

119894=1119899119886119894

119872(Δ119910120575119910)minus119896

sum

119894=1

119899119886119894sdot (119899

119887(119894+119896)minus 120575

1198861198871205751198960)

119861119870(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus 1198861198871

sum119872

119894=1119899119886119894

119872(Δ119910120575119910)

sum

119894=1minus119896

119899119886119894sdot (119899

119887(119894+119896)minus 120575

1198861198871205751198960)

(16)

where 119896 isin [0119872 minus 1] and delta functions indicate thecancellation of self-pair distributions

(iii) Conditional probabilities the single- and two-pointprobabilities can be given in terms of the jointmultiplicity

119875119894(119886) =

119899119886119894(119899

119887minus 120575

119886119887)

119873119886(119873

119887minus 120575

119886119887) 119875

119894119895(119886119887) =

119899119886119894(119899

119887119895minus 120575

119886119887120575119894119895)

119873119886(119873

119887minus 120575

119886119887)

(17)

In statistics and probability theory the Bayes the-orem shows the importance of the mathematicalmanipulation of the conditional probabilities TheBayesian probability is one of different interpretationsof probability and belongs to evidential probabilitiesIn an ensemble the Bayes theorem gives

1198751198951(119886119887 Δ119910 120575119910) equiv

119875119894119895(119886119887)

119875119894(119886)

=119899119886119894(119899

119887119895minus 120575

119886119887120575119894119895)

119899119886119894(119873

119887minus 120575

119886119887) (18)

This is the conditional probability that predicted thata particle with charge 119887 occupies the 119894th bin while the119895th bin is occupied by another particle with charge119886 as determined by the joint distribution 119899

119886119894(119899

119887119895minus

120575119886119887120575119894119895)

Regarding balance functions the conditional proba-bility is defined as

119875119896Δ119910(119886119887) equiv

119873119896(119886119887 Δ119910 120575119910)

119873119886(Δ119910) (119873

119887(Δ119910) minus 120575

119886119887)

119861119896(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus119886119887119875

119896Δ119910(119886119887)119873

119886(Δ119910) (119873

119887(Δ119910) minus 120575

119886119887)

119873119886(Δ119910)

119861119896(Δ119910 120575119910)

equiv1

2sum

119886

119875119896Δ119910(119886119886) minus

sum119886119887119875119896Δ119910(119886119887)119873

119886(Δ119910)119873

119887(Δ119910)

119873119886(Δ119910)

(19)

where sum119896119875119896Δ119910(119886119887 Δ119910 120575119910) equiv 1


times10minus3

1

05

0

minus05

minus1

Most central ()70 60 50 40 30 20 10 0

STAR 200GeVSame charge AuAuOpp charge AuAu

Same charge CuCuOpp charge CuCu

⟨cos

(120601120572+120601120573minus2Ψ

RP)⟩

Figure 2 Angular correlations as measured by STAR for Au+Auand Cu+Cu collisions The shaded areas stand for systematicuncertainties in the analysis relative to the elliptic flow The figureis taken from [29]

21 Angular Correlation For odd-parity observables in STARexperiment at RHIC large fluctuations have been observed[52 53] These fluctuations are supposed to arise from thecolor flux tubes which carry both kinds of color chargesthat is color-electric and color-magnetic flux The color fluxtubes generate electric field with random signs [29] Theelectric field fluctuates as 1(radic119873fluxtubes) where 119873fluxtubes isthe number of tubes The correlation between positive andnegative charges are conjectured to includ large fluctuationsfrom odd-parity Obviously both types of charges should beproduced at same space-time coordinates In other wordsboth charges should have the same rapidity and azimuthalangle in the collective flow Such correlations can be describedby BF The correlations can be expressed as ⟨cos(Δ120601balance)⟩[29]

120574+minus= 119865

119876((sum

119894

cos 2120601119894⟨cos (Δ120601balance)⟩ (120601119894)

minus sin 2120601119894⟨sin (Δ120601balance)⟩ (120601119894))

sdot (119872+)minus1

)

(20)

where Δ120601balance = 120601119895 minus 120601119894 and 119865119876 is the fraction of chargeMomentum conservation means sum

119894119901119894

119909= 0 sum

119894119901119894

119910= 0

The correlations are shown in Figure 2 in dependence on thecollision centrality

When themomentum119901119894119909= 119901

119894

119905cos(120601

119894) the correlation can

be written as [29]

120574 = minus119865119901

sum119894(cos2120601

119894minus sin2120601

119894)

1198722

tot (21)

Here 119865119901is fraction of the momentum balance and 119872tot =

119872++ 119872

minus+ 119872

0sums over positive negative and neutral

charges The fluctuations are essential in estimating theelectric field in the initial conditions which is found 10of the magnetic field Thus the charge and momentumconservation should be attributed to the correlation withone unit of rapidity while the fluctuations for the initialconditions are found with several units of rapidity

22 Advantages of Balance Functions In light of the variousdefinitions of BF Section 2 different advantages can be listedout(i) Charge-Density Balance Instead of determining the net-charge density it is advantageous to study the associatedcharge density balance [17](ii) Associated Charge-Density Distributions The charge-density balance allows us to select out the associated chargedensity distributions and the correlated fractions [17] Theassociated charge-density balance has a further advantageThis is less sensitive to the acceptance corrections than theassociated charge density itself Taking the trigger of a largetransverse momentum event as the selected particle(s) thedependence of the associated charge-density balance Δ119902 onthe rapidity of other particles was presented in [17](iii) Relative Distance The balance functions are able to mea-sure the relative distance between the positive and negativecharges produced in heavy-ion collisions In the same waythey can be applied to the baryon and antibaryons and soforth(iv) Charge Fluctuation The charge fluctuations which occurin heavy-ion collisions are related to the charge-balancefunctions So that it is very important to study the evolutionof state of matter created during the collision This can bedone by calculating the charge correlations in dependence onthe rapidity(v) Width of Balance Functions The production time ofthe positive-negative pair of charges can be determined bystudyingwidths of BF in terms of the rapidity [38] It is arguedthat narrowed balance functions are considered as probes ofdelayed hadronization due to the reduction in the diffusiontime This implies long-lived stage before hadronization Inother words this might refer to delayed hadronization [54](vi) Rapidity Correlation One of the most important featuresof the balance functions is the boost invariance variable suchas rapidity The rapidity correlations describe what so-calledthe conditional probability This estimates the probabilityof the charge produced in a rapidity bin associated to theopposite charge in the other rapidity bin Rapidity and


0 05 10

01

02

03

04

05

06

07

minus05 lt y lt 05

0 lt y lt 1

1 lt y lt 2

15 lt y lt 25

B(Δ

y|yw)

Δy

(a)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

B(Δ

y|yw)

Δy

(b)

Bs(Δy)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

Δy

(c)

Figure 3 The 119901119879-integrated 119861(Δ119910 | 119910

119908) of final hadron system at different rapidity positions with same (a) and different (b) window sizes

as well as the 119861119904(Δ119910) (c) Correlation coefficient 120588 is taken to be 03 The graph is taken from [30]

pseudorapidity were given in (1) and (2) respectively Bothact as measure for the speed(vii) Probing Hadron- and QGP-Formation One of the signa-tures for theQGP formation is the sudden drop in the balancefunction width [55] On the other hand having an access tothe occurrence of quark-pairs can be utilized as a signaturefor the hadron formation or hadron diffusion

23 Short-Comings of Balance Functions The balance func-tions can have some short-comings(i) Binning Geometry and Bayes Theorem The conditionalprobability is not a true probability Using it leads to con-tradiction between the binning geometry and Bayes theorem[49](ii) Nonstandard Normalization The normalization of BF isnot standard one [49](iii) Length Scale Inconsistency It is argued that in nucleus-nucleus collisions the production of pair separation lengthat the formation stage is zero [15] This is not compatiblewith the fragmentation scenario [49] In the thermal anddiffusion process of elementary particle collisions the hadrondiffusion is negligible while the correlation length that wouldbe charge-dependent is larger [49]

3 Experimental Measurements

The experimental features of NA22 [56] and STAR experi-ments [57]were essential to enable both of themanalyzing thecharacteristics of BF [15 28] which can be used as effectiveprobes for the phase transition in heavy-ion collisions ande+ + eminus collisions at ISR and PETRA energies [58] Manymeasurements for the dependence of BF on the collisioncentrality [35] the system size [25ndash27] and the transversemomentum [57] have been conducted All properties men-tioned above which can be categorized under what so-called

the longitudinal boost invariance are very useful in studyingBFTheboost invariancemeans that the single particle densitywill be independent of the rapidityTherefore it is essential tostudy BF in terms of rapidity in order to investigate the boostinvariance The widths of balance functions get narrowerby increasing the window size 119910

119908[30] This relation can be

formulated from the following relation

119861 (Δ119910 | 119910119908) = 119861 (Δ119910 | infin) (1 minus Δ119910) (22)

31 Various Measurements One can categorize the exper-imental measurements [54] according to the type of thereaction and the dependence of the quantities of commoninterest

(i) The type of the reaction whether nuclei hadron orhadron-nuclei interaction the hadron-hadron colli-sions like positive pion and kaon 120587+119901 119896+119901 atradic119904NN =22GeV in NA22 experiment were introduced in [56]This experiment can compromise the full momentumand 4120587 azimuthal acceptance so that one can verywell determine the properties of BF

(ii) The dependence on the rapidity (pseudorapidity) andthe window size the window size can be arbitrary butit should be restricted by the rapidity range Figure 3shows BF in terms of the rapidity positions and atdifferent window sizes [30]

(iii) Multiplicity dependence it is found that as the sys-tem size becomes large (in central collisions) mostof QGP signatures can be observed [28] Due tothe difficulty of the experimental determination ofthe collision centrality we are left with the Monte-Carlo simulations to play this role Therefore themultiplicity of observed particles can be correlatedto the collision centrality [28] The balance functionsare integrated for all events (multiplicities) in thepp collisions and plotted in Figure 4 which shows


Pions

0

02

04

06 K0s

1205880

0 04 08 12 16

B(q

inv)

((G

eVc

)minus1)

qinv (GeVc)

(a)

Kaons02

01

0

0 04 08 12 16

120593

qinv (GeVc)

B(q

inv)

((G

eVc

)minus1)

(b)

Figure 4 The balance functions are given in terms of 119902inv for charged pion pairs in panel (a) and charged kaon pairs in panel (b) from ppcollisions atradic119904NN = 200GeV integrated over all multiplicities The graph is taken from [28]

the dependence on the 119902inv integrated over all mul-tiplicities atradic119904NN = 200GeV [28]

(iv) Beam energy dependence Figure 5 shows the depen-dence of BF on the center-of-mass energy radic119904NNranging from 77 to 200GeV [28] The figure showsthe relation between BF and pseudorapidity for themost central collisions 0ndash5 It is to be noticed thatBF behave as well at different energies The data fromSTAR is narrower than the shuffled results

(v) Correlation the balance functions of charge correla-tions and fluctuations depend on the charges square[56 59 60]

⟨(1205751198762

)⟩ = ⟨1198762

⟩ minus ⟨119876⟩2

= 1199022

(⟨1198732

⟩ minus ⟨119873⟩2

) (23)

where119876 = 119899+minus119899

minusand119873ch = 119899+ +119899minus For hadron gas

119902 = plusmn1 while 119902 = plusmn13 plusmn23 for QGPFurthermore

⟨119873ch⟩ ⟨1205751198772

⟩ = 4⟨(120575119876

2)⟩

⟨119873ch⟩ (24)

where

119877 =⟨119873

+119873

minus⟩ minus ⟨119873

minus⟩ ⟨119873

+⟩

⟨119873minus⟩ ⟨119873

+⟩

(25)

Then the119863-measure for fluctuation can be written as

119863 (119876) = 4⟨(120575119876)

2

⟩

119873ch (26)

The correlations of all charges are conjectured tocombine with BF

119863 (119876)

4= 1 minus int

119910119908

0

119861 (Δ119910 | 119910119908) 119889Δ119910 +

⟨119876⟩

119873ch (27)

(vi) Centrality dependence BF have been studied atdifferent collision centralities and noticed that theycoincide but the width changes due to the differentpositions of the rapidity ranges minus05 lt 119910 lt 050 lt 119910 lt 1 1 lt 119910 lt 2 and 15 lt 119910 lt 25 [28] Shuffleddata and mixed collisions are analyzed as well Formixed collisions the balance functions are zero at allthe nine centrality bins Figure 6

(vii) Transverse momentum dependence BF can also bestudied in terms of the difference ofmomenta (invari-ant) of the produced particles that is 119902inv In aGaussian-like form

119861 (119902inv) = 1198861199022

inv119890minus1199022

inv21205902

(28)

This was implemented for charged kaons 119870plusmn fromAu+Au collisions at radic119878NN = 200GeV in differentcentrality bins The mixed events were abstractedfrom these balance functionsThe solid curves are theone calculated from (28) In [28] the authors statedthat the peaks observed in each curve are due to thedecay of 120601 rarr 119896

++119896

minus Figure 7 shows these relations

32 Confronting to STAR Experiments Measuring BF datesback to 2003 where the STAR experiment announced its firstmeasurements [25]

321 System Size and Centrality Dependence The balancefunctions were measured in various system sizes for exampleAu+Au at radic119904NN = 200GeV in the STAR experiment [54]and Pb+Pb collisions at radic119904NN = 172GeV in the ALICEexperiment at LHC [37] Also the width of BF was measuredin Pb+Pb C+C and Si+Si collisions at radic119904NN = 88 172 GeVat SPS [16] It was observed that BF behave as well in boththe central and peripheral collisions but the widths changeThis behavior was investigated at different pseudorapiditywindows [54] The width of BF is considered as a timometerfor the hadronization It was observed that the narrowing of


77GeV 196GeV

27GeV 39GeV 624GeV

115GeV

B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06

DataShuffled

200GeV

Δ120578

B(Δ

120578)

0

02

04

06

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18

Figure 5 The balance functions in terms of Δ120578 for all charged particles Central events (0ndash5) are shown here at radic119904NN ranging from 77 to200GeV The graph is taken from [28]

BF in central collisions is more than in peripheral collisions[37] and this agrees well with the theoretical results [37] forlate hadronization or long-lived QGP In Au+Au collisionsat radic119904NN = 200GeV it was concluded that increasing thecentrality and the transverse momentum decreases the widthof BF [54] due to the radial flow [54] The dependence ofbalance functions ⟨Δ120578⟩ on the mean number of woundednucleons was studied [27] A strong centrality dependencewas found in pp collisions and width of ⟨Δ120578⟩ decreases withincreasing centrality of Pb+Pb collisions [27]

322 Chemical Evolution of QGP In heavy-ion collisions itis conjectured that the creation of quarks occurs in specificspace-time while the antiquarks may occupy the samecoordinates [33] This would mean that the charge balancefunctions can identify the location of the balancing for theproduced hadron [55] Then the rapidity distribution of thebalancing charges can be observed for any pair flavors [55]

Therefore the charge correlation function can be analysedeven in the QGP medium [55] Obviously BF can be relatedto the correlation function [55] In order to determine BF fordifferent particle species (hadrons) the longitudinal positionin the Bjorken coordinates in which the charge densityis depending should be analyzed [33] The correlationsfrom charge conservation should be affected by the timeof creation of charge-anticharge pairs [47] By analysingcorrelations from STAR experiment for different particlespecies Pratt [55] distinguished the two separate waves ofcharge creation expected in high-energy collisions one atearly times when the QGP should be formed and a second athadronization Further the density of up down and strangequarks was extracted in QGP and found in agreement withpredictions for a chemically thermalized plasma (at a level of20)

In relativistic heavy-ion collisions thousands of hadronsare created For every quark flavor detected in the final state


DataShuffled

DataShuffled

DataShuffled

B(Δ

120601)

Δ120601

39GeVAll charged particles

0ndash5 5ndash10 10ndash20



0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

0 1 2Δ120601

0 1 2Δ120601

0 1 2 3

3

3

3

3

3

3

3

3Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2

Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2

Figure 6 The balance functions in terms of Δ120601 for all charged particle pairs from Au+Au collisions at radic119904NN = 39GeV The graph is takenfrom [28]

like 119906 119889 and 119904 quarks there should be antiquarks 119906 119889 and 119904too Such quark correlations are defined as [33]

120594119886119887=⟨119876

119886119876119887⟩

119881 (29)

where 119876119886is the net-charge of 119906 119889 and 119904 quarks within the

volume 119881 For a parton gas

120594QGP119886119887

= Δ119886119887(119899

119886+ 119899

119886) (30)

where 119899119886 119899

119886are densities for 119906 and 119889 quarks and their

antiquarks respectively For a noninteracting hadron gas thecorrelation is defined as

120594HG119886119887= sum

120572

119899120572119902120572119886119902120572119887 (31)

where 119902120572119886

is the charge of type 119886 and 120572 is the particle typeThe correlations for different specieswere calculated by latticegauge theory [31 32] Figure 8

The correlation of hadrons is given as [33]

119866120572120573(120578) = 4sum

119886119887119888119889

⟨119899120572⟩ 119902

120572119886120594(had)(minus1)119886119887

(0) 119892(had)119887119888

sdot (120578) 120594(had)(minus1)119888119889

(120578) 119902120573119889⟨119899

120573⟩

(32)

The balance functions should be related to that correlation

119861120572120573(Δ120578) =

119866120572120573(Δ120578)

119899120573+ 119899

120573

(33)

where 120573 is the hadron species and 119899120573is the number per

rapidity of that species Therefore BF for identified pair ofspecies can be calculated [33]

323 Dependence on Beam Energy and Reaction PlaneInformation on the creation of hot and dense matter can beextracted by studying the correlations and fluctuations [34]


DataShuffled

DataShuffled

DataShuffled

Kaons

B(q

inv)

1205942ndf = 174838

120590 = 0501

1205942ndf = 807638

120590 = 0504

1205942ndf = 65438

120590 = 0518

1205942ndf = 251638

120590 = 0496

1205942ndf = 753538

120590 = 0509

1205942ndf = 680438

120590 = 0526

1205942ndf = 482238

120590 = 0503

1205942ndf = 963838

120590 = 0519

1205942ndf = 445938

120590 = 0530




03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1 2

2

2

2

2

2

2

2

2

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

Figure 7The balance functions in terms of 119902inv for charged kaon pairs fromAu+Au collisions atradic119904NN = 200GeV in different centrality binsSolid lines correspond to (28) The graph is taken from [28]

The balance functions can directly measure the correlationsbetween negative and positive charge pairs [34] They aresensitive to the changes in the formation or diffusion pro-cesses of the balancing charges [34] If the hadronizationprocess delays the particle and antiparticle are correlated dueto the conservation of the charge [34] In addition to that thereaction plane would play a vital role as BF depend on theazimuthal angle

119861 (120601 Δ120601) =1

2

Δ+minus(120601 Δ120601) minus Δ

++120601 Δ120601

119873+(120601)

+Δ

minus+(120601 Δ120601) minus Δ

minusminus120601 Δ120601

119873minus(120601)

(34)

where 119873+(minus)(120601) is the total number of +ve and (minusve) par-

ticles Δ+minus(120601 Δ120601) is total number of positive particles with

azimuthal angle 120601 with respect to the reaction plane and the

negative particles with Δ120601 with respect to the positive one[34] The width of BF is given as

⟨Δ120578⟩ =sum

119894119861 (Δ120578

119894) Δ120578

119894

sum119894119861 (Δ120578

119894) (35)

Figure 9 shows the widths of BF in terms of the pseu-dorapidity Δ120578 and azimuthal angle Δ120601 in dependenceon the participant particles and the center-of-mass energyrespectively The calculations are compared with the STARdata for the most central events (0ndash5) of Au+Au collisionsatradic119904NN = 200 624 39 115 and 77GeV It can be concludedthat the narrower width indicates an early hadronizationtimewhile awider one indicates the diffusion after the freeze-out [34] Also it is noticed that the dependence of identifiedkaons on the centrality is weak in contrast to the pions [34]indicating that the kaons are likely produced in very earlystage of the collision


0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs

Figure 8 The charge fluctuations as functions of temperature inthe lattice gauge theory [31 32] Results at temperatures lt160MeVare likely belonging to hadronic state At higher temperatures thesystem is characterized by partonic degrees of freedom The graphis taken from [33]

33 Confronting to ALICE Experiment

331 Energy Dependence When comparing the results givenin [35 37] with each other one finds that in [37] thewidth of the balance functions is studied in terms of thepseudorapidity ⟨Δ120578⟩ and ⟨Δ120601⟩ For a better comparisonwith STAR results ALICE measurements were corrected foracceptance and detector effects So that terms119861

+minus(Δ120578 | 120578max)

should be corrected

119861+minus(Δ120578 | 120578max) = 119861+minus (Δ120578 | infin)(1 minus

Δ120578

120578max) (36)

It is obvious that the BF width is narrower at LHC than atRHIC energies Figure 10

On the other hand Figure 11 represents ⟨Δ120578⟩ and ⟨Δ120601⟩as function of the average number of participant particlesfrom peripheral to central collisions The dependence on thenumber of participants is appropriate choice for scaling to thecentrality classes

4 Effective Model Calculations

41 Coalescence Model One of the strongest signatures forQGP [61] is the suppression of charmonium system 119869120595 asmeasured in Pb+Pb collisions [62] The quark coalescencefrom deconfined quarks to produce charmed hadrons canbe best described by the algebraic coalescence model forrehadronization of charmed quark matter (ALCOR) Thenumber of produced hadrons is given by the number ofquarks or antiquarks which mainly are the compositionsof those hadrons multiplied by the coalescence coefficient119862119902and the nonlinear normalization coefficient 119887

119902 in which

the latter indicates the conservation of the quark numberduring the quark coalescence [63]TheALCORmodel beginswith the valence quarks and antiquarks that create the finalhadron-state in thermal equilibrium [64] In the ALCORmodel meson and baryon coalescence coefficients are repre-sented by 119862

119872(119894 119895) and 119862

119861(119894 119895 119896) respectively where 119894 119895 and

119896 refer to the quark species numbers Also a normalizationfactor and spin degeneracy factor 119863ℎ

= 2119878ℎ+ 1 can be

introduced in this model where 119878ℎis the hadron spin Thus

the number of a certain type of meson that has flavors 119894 and 119895is given as [64]

119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895

are the number of quarks and antiquarks[65] and 119887

119902119894

and 119887119902119895

are the corresponding parametersrespectively The number of a certain baryon with flavors 119894119895 and 119896 is given by

119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)

are the number of quarks and antiquarksof type 119894 for instance One can reformulate (39) as sum over119894 119895 and 119896 for each hadron from 1 to 119899

119891flavors So that in

ALCORmodel one can calculate the hadron multiplicity andcompare between themodel and the experimental results [6566]

Changing linear to nonlinear rehadronization coales-cence model is doable The linear coalescence model isbased on the counting of quarks and the determination ofprobabilities in the heavy-ion collisions It was assumed [67]that the number of produced particles is directly proportionalto the product of constituent quarks in the reaction volume[68]

119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)

The antiparticles are straightforwardly constructed [68]

119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)

The coalescence model can be used to predict the smallwidth of the baryon-antibaryon BF [50] It is observed that


77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)

Figure 9 The widths of balance functions are given as functions of Δ120578 Δ120601 for all charged particle pairs from Au+Au collisions Left-handpanel shows centrality dependence while the right-hand panel shows beam energy dependence of most central events (0ndash5) The graph istaken from [34]

in the central heavy-ion collision at RHIC energies [25]the hadron constituents of quarks which are described bycoalescence model [63] can explain the small pseudorapiditywidth of BF Furthermore the coalescence concept wouldexplain cluster from pairs of charges

119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)

For the above processes the momentum distribution for thetwo particles can be written as [50]

120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875

2are the momenta of the two clusters The

momenta of quarks and antiquarks are 119901 and 119901 respectively120588119888(119875) is the distribution of clusters and 119891(119875 119901) and 119891(119875 119901)

are the cluster dissociation probabilities of finding a quarkor antiquark of momentum 119901 andor 119901 in the clusterrespectively119866(119875minus119901) is the coalescence probability in whichthe quark-antiquark pair coalesce to create a hadron


80

⟨Δ120578⟩

07

06

05

Centrality percentile0 20 40 60

STAR Au-AuradicsNN = 200 GeVALICE = 276 TeVPb-Pb radicsNN

(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)

Figure 10 From ALICE and STAR experiments the centrality dependence of the balance function width ⟨Δ120578⟩ (a) and ⟨Δ120593⟩ (b) The STARresults [35] have been corrected for the finite acceptance as suggested in [36] The figure is taken from [37]

⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400

Pb-Pb at radicsNN = 276 TeV

Pb-Pb at radicsNN = 172 GeVAu-Au at radicsNN = 200 GeV

(a)

Pb-Pb at radicsNN = 276 TeVAu-Au at radicsNN = 200 GeVPb-Pb at radicsNN = 172 GeV

⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)

Figure 11 The centrality dependence of the relative decrease of BF width in relative pseudorapidity (a) and relative azimuthal angle (b) TheALICE results are compared with the results for the highest SPS [26 27] and RHIC [35] energies The figure is taken from [37]

Similarity the distribution of baryon and antibaryonldquothree particlesrdquo distribution can be written as

120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)


which is valid for each quark and antiquark [64] This sumsover the different number of flavors so that the number ofquarks and antiquarks of type 119894 is given by 119873

119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)

The calculation of BF in the coalescence model hasthe ability to explain the small pseudorapidity width of BFobserved for central heavy-ion collisions [63] where theparameter 120572 = 1198882ℎ2 For uncorrelated decay 119888ℎ ≃ 0

42 Thermal Resonances As discussed in previous sectionsthe STAR analysis of balance functions is based on multiplic-ities [25]

119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873

minus+(Δ) minus ⟨119873

minusminus(Δ)⟩

119873minus

(46)

where119873+minus(Δ) counts the opposite-charge pairs having rapid-

ity 119884 relative to |1199102minus 119910

1| = Δ at 119884 sim 119884

max and BF ofall changed hadrons are normalized to unity The separationof balancing charges at kinetic freeze-out is studied [69] Tocharacterize the possible contributions we highlight that the120587+120587minus BF have two types of contributions corresponding to

two different mechanisms of their creation The resonancesmay come up with an additional contribution The decaychannels of neutral hadronic resonances likely lead to 120587+ minus120587minus pairs Also a nonresonance contribution is related to

other correlations among the charged particles The twoopposite-charge particles are produced at the same space-time coordinates with thermal velocities A neutral resonanceends up as a 120587+ minus 120587minus pair where as in the nonresonancemechanism of charge balancing a charged pion can bebalanced with another charged hadron not necessarily apion [38] In light of this the 120587+120587minus balance functions canconstructed as

119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)

The resonance contribution 119861119877(Δ 119884) is obtained from the

expressions describing the phase-space of the pions emittedin a decay [38] The calculation in the neutral clusters model[63] does not depend on the correlations between the clustersthemselves But they are determined by the single-particledistribution or by two-particle distribution in which the pairof particles can be formed from one cluster and others fromdifferent clusters [63] Replacing the neutral clusters by theneutral resonances in order to obtain the two-particle rapiditydistribution of the 120587+ minus 120587minus pairs stemming from the decay ofa neutral resonance then the two-particle pion momentumdistribution in two-body 120587+ minus 120587minus resonance decay can beexpressed by Dirac 120575 function

120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901

2are total momentum momentum of

positive pion and momentum of negative pion respectivelyand the 119887

120587120587is the branching ratio The normalization factor

1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)

The correlation between nonresonance pions is not specifiedby the model introduced in [38] It is assumed that thecreation of an opposite pair occurs in the fireball cylinderthat is the two charges have the same longitudinal andtransverse collective velocity [38] The results are shownin Figure 12 The calculations for four different centralitywindows are compared to the STAR data [25]

43 Statistical and Dynamical Model At top RHIC energiesan energy density can be as high as ≃10GeVfm3 Apparentlythis would cover a volume of several hundred fm3 in theAu+Au collisions [25] Therefore quark and gluon degreesof freedom provide a description of the microscopic motionfor several fmc until the matter expands and cools downtill the hadronic degrees of freedom become appropriate [39]The conversion frompartonic to hadronic degrees of freedomaccompanied by increasing production of quark antiquarkpairs on the entropy stored in gluons and quarks is convertedto hadrons each of which has at least two quark The changein the degrees of freedom accompanying the hadron-quarkphase transition was revised in [70ndash75] There newly createdcharges are more correlated to their anticharges than pairscreated early [39]

119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875

2are ldquothe extra particle of the opposite charge

with momentum 1198752given the observation of the first particle

with momentum 1198751rdquo as stated in [39] and +minus indices refer to

particles or antiparticles respectively The balance functions


05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)

Figure 12 The balance functions for pions in the thermal model calculated for four different centralities are compared to data [25] 120575 equiv ΔThe graph is taken from [38]

are designed as measure for the probability of observing anextra particle with opposite charge and momentum 119875

2gives

the observation of the first particle with momentum 1198751 119875

1

refers to a particle observed anywhere in the detector and1198752refers to either the relative rapidity Δ119910 or the relative

momentum 119876inv The STAR measurements were performedfor all charged particles as functions of relative pseudorapid-ity and for identical poins as functions of relative rapidity[25] The behavior of the balance function is comparedbetween the STAR data [25] and the one calculated fromthe microscopic hadronic simulations RQMD (relativisticquantum molecular dynamic) [76] Figure 13 has shown the120587+120587minus balance functions from RQMD for p+p and Au+Au

collisions compared to the STAR data [25]

44 Thermal Blast-Wave Model The dynamical evolution ofthe system created in heavy-ion collisions can also be studiedin the blast-wave model [77] which describes the kineticfreeze-out properties in which the particles are thermalizedat the kinetic freeze-out temperature [28] The creation ofparticles in a very hot and dense matter has the features ofexplosion [78] The explosion wave called blast wave dueto sequential collisions The hot and dense medium wouldbe anisotropic so that the velocity of the particles is also

anisotropic [78] Finally the net-flow of velocity 120573 can beestimated [78] The model has eight parameters 119877

119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877

119910 and 119879 are the radii of

the transverse shape and the temperature respectively 119886119904is

the surface diffuseness parameters 1205880and 120588

2are the radial

and ansiotoropy flow parameters respectivelyThe schematicdiagram Figure 14 shows the elliptic flow with 119877

119909and 119877

119910

[40]In principal the thermal models can divide the balancing

charges into resonant and nonresonant contributions [38]The resonant contribution is dominated by the decays of thehadron resonances to create 120587+120587minus in the most final state[38] while nonresonant contribution is dominated by otherprocess or correlations between charges Accordingly BF canbe expressed as [38]

119861 (Δ119910 119910119908) = 119861resonant (Δ119910 119910119908) + 119861non-resonant (Δ119910 119910119908)

(51)

where Δ119910 = 1199101minus 119910

2and 119910

119908is the window size ranging

from 1 to 4The resonant contribution can be estimated fromthe cluster model [63] While the nonresonant contributioncan not be determined specifically Bozek et al [38] proposeda form in which the charge-anticharge pair is created in afireball cylinder [38] BF calculated due to resonance and

16 Advances in High Energy PhysicsB(Δ

y)

0 1 2 3 4 5 6

Δy

Au+Au RQMD 0 lt b lt 5Au+Au RQMD 5 lt b lt 10

Au+Au RQMD 10 lt b lt 14p+p RQMD 0 lt b lt 5

05

04

03

02

01

0

Figure 13 120587+120587minus balance functions for RQMD are shown for bothp+p and Au+Au collisions assuming a perfect detector In contrastto the experimental results of [25] the balance functions are slightlybroader for central Au+Au collisions The graph is taken from [39]

Eventplane

120601b

120601s

Figure 14 The elliptical subshell of the source Here 119877119910119877119909are radii

of the ellipse The arrows represent the direction and magnitude ofthe elliptic flow The graph is taken from [40]

nonresonance contributions [38] replace the neutral cluster[63] by neutral resonances Then the two-particle rapiditydistribution for pair for instance pion pair is obtained

119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp

sdot int 119889119901perp

1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)

The nonresonant rapidity distribution is given as

119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions

Pions fromresonances

05 1 15 2 25

Figure 15 Resonance and nonresonance balance functions plottedas a function of the rapidity difference 120575 refers to Δ119910 Δ119910 equiv 120575119910 Thegraph is taken from [38]

From (52) and (53) the resonance and nonresonance BFfor pion pairs can be calculated

119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873

120587minus)2 The resonance and nonreso-

nance balance functions are given in Figure 15In heavy-ion collisions the quarks and gluons are under

collective expansion that is geometric asymmetry of planeof the interaction can be studied as anisotropic flow while thesecond coefficient is called the elliptic flow [40]These contri-butions are Fourier expansion of the differential distribution

1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)

The Fourier decomposition is given as [79]

1 + 2V1cos (120601 minus ΨPR) + 2V2 cos (2 (120601 minus ΨPR)) (56)

where V1is the directed flow V

2is the elliptic flow and ΨPR is

the real reaction plane [79]The elliptic flow is essential probeto studying the evolution of the strongly interacting systemand the flow fluctuations and balancing between createdcharges [80ndash82]

An extended blast wave model was introduced in orderto investigate the effect of flow in which a combination ofelliptic flow with the transverse mass spectra and the two-charge correlationwas introduced [79]This blast wavemodeldescribes a specific particle elliptic flow that emitted throughan finite thin shell In order to determine the size of pionsproduced in the reaction the model has to be extended


00

002

004

006

008

01

012

014

016

B(Δ

120601)

minus150 minus100 minus50 50 100 150

Δ120601

Data minus75∘ lt 120601 lt 75∘

Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘

Model minus75∘ lt 120601 lt 75∘

Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘

Figure 16 The balance functions for 120601 = 0∘ (in-plane) 120601 = 45∘and 120601 = 90∘ (out-of-plane) particles pairs The 40ndash50 centralitybins are shownThe points are from the data (not corrected for eventplane resolution) while solid lines represent the blast-wave modelcalculations The graph is taken from [34]

through a filled cylinder The significant idea of the extendedblast-wave model is to describe the system in the freeze-outconditions in terms of the elliptic flow and temperature [83]Some new parameters concerning the geometry of the systemwere introduced as well [84 85] The new parameterizationinterprets the transverse mass spectra as mentioned aboveThe probabilities of emitting particles in the space-time 119883with momentum 119875 can be written as [83]

119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)

where Θ is the step function modelling the confinement ofthe system in the filled ellipse The spatial and azimuthalmomentum are 120601

119904and 120601

119901 respectively The earlier gives the

radii of the system in-plane while the latter gives the out-of-plane Figure 16 shows BF calculated in the blast-wave modelcompared with STAR data at different azimuthal angles [34]while Figure 17 shows the blast-wave model calculationscompared with midcentral peripheral and central collisionsfrom STAR data [25]

For completeness we add that the evolution of the systemtill the final state would be more convenient to be studiedby the Hanbury Brown-Twiss (HBT) interferometry [86ndash88]In that case measured single- and two-particle correlationsare essential inputs [89 90] The probability for a jointobservation of the two quanta with momenta 119896

1and 119896

2and

the correlation function are also studied [89]

04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)

PeripheralMidcentral

CentralBlast wave

Figure 17 The balance functions from 200119860GeV Au+Au collisionsmeasured by STAR are compared to the canonical blast-wave modeldescribed in the text The model should set a lower bound for thewidth of a balance function provided that the particles are emittedthermally The remarkable agreement with the data suggests thatcharge conservation remains highly localized at breakup The graphis taken from [39]

45 Glue Cluster Model The experimental results forinstance from STAR [25 35 91] and NA49 [26 27] shouldbe understood that the charges are produced in a late stage ofthe hadronization process that is in freeze-out region [92]This means that QGPmostly consisted of gluons as wellThewidths of BF in the central and peripheral collisions are dif-ferent and also they are different fromAA and pp collisions Itis argued that the systemwould needmore correlations in theQGP phase exhibiting a clustering behavior So that the glueclusters can explain the correlations in QGP In momentumspace the width of BF can be determined by the short-rangecorrelations as proposed by the STAR experiment [35] Itis believed that the small or narrow width of BF indicatehow late is the stage of hadronization Apparently this wasalso measured by the STAR experiment and expected fromdifferent models like the coalescence model The clustersdecay to gluons and quark-antiquark pair for instance to upand antiup quarks Both quarks should attempt to recombineagain forming pions or any other kind of mesonsThe clusterdecay distribution is given by

120588 (120578) =1

2 (cosh 120578)2 (58)

The decay width ⟨|120578|⟩ = log 2 Thus the width of BF can beaffected also by the transverse flow The clusters are isotropicin their rest frame However after the transverse flow ofclusters they become no longer isotropic

46 UrQMD The ultrarelativistic quantum moleculardynamics (UrQMD) model is a microscopic model used to


Au+Au UrQMDAu+AuAu+Au shuffledp+pd+Au

Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩

Pions Au+AuKaons Au+AuPions Au+Au HIJINGKaons Au+Au HIJINGPions Au+Au UrQMDKaons Au+Au UrQMD

Pions Au+Au blast wave

Pions p+pKaons p+pPions p+p HIJINGKaons p+p HIJING

(b)

Figure 18 (a)The balance function width ⟨Δ120578⟩ for all charged particles fromAu+Au collisions atradic119904NN = 200GeV compared with the widthsof BF calculated using shuffled events The balance function widths for p+p and d+Au collisions atradic119904NN = 200GeV are also shown FilteredUrQMD and HIJING calculations are shown for the widths of BF from Au+Au collisions (b) The same as in (a) but for identified chargedpions and charged kaons The width of BF for pions predicted by the blast-wave model [39] is also shown The figure is taken from [28]

simulate (ultra)relativistic heavy-ion collisions in the energyrange from Bevalac to LHC Main goals are to gain betterunderstanding about the following physical phenomenawithin a single transport model

(i) creation of dense hadronic matter at high tempera-tures

(ii) properties of nuclear matter delta and resonancematter

(iii) creation of mesonic matter and of antimatter(iv) creation and transport of rare particles in hadronic

matter(v) creationmodification and destruction of strangeness

in matter(vi) emission of electromagnetic probes

Figures 18 and 19 show the balance function widths forpions and kaons and also the widths in terms of 119902long 119902sideand 119902out respectively All are compared to the STAR data forAu+Au collision at 200GeV Filtered HIJING calculationsSection 47 are also shown for the widths of BF from pp andAu+Au collisions

47 HIJING The heavy ion jet interaction generator(HIJING) was developed by Gyulassy and Wang [93] with

special emphasis on the role of minijets in proton-protonproton-nucleus and nucleus-nucleus interactions at colliderenergiesThe perturbative QCD predicts jet production fromparton scatterings in high energy hadronic interactions It istherefore expected that hard or semihard parton scatteringswith transverse momentum of a few GeV are expected todominate high energy heavy ion collisions The HIJINGcode has been widely distributed to experimental groupspreparing for RHIC and LHC HIJING is also used toinvestigate two effects gluon shadowing and jet quenchingin heavy ion collisions at RHIC [42] The study of pA andAA collisions is required to separate between the two effectsat RHIC Therefore the conclusions from such study willinvestigate the new physics of the gluon structure of nucleiand the energy loss in QGP As introduced the BF width inthe rapidity representation can be defined as

⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)

HIJING can establish the existence of QGP by thesimulation and extractingBF ButHIJING lacks the collectiveflow description so that generation of the balance functionwidths by HIJING is larger than that measured in exper-iments Figure 20 represents the balance function widthsfrom HIJING and the multitransport (AMPT) model with


Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000

Au+Aup+pHIJING Au+Au

120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart

Blast waveUrQMD Au+Au


radic2mKTkin

UrQMD Au+Au

Figure 19 The balance function width 120590 extracted from 119861(119902inv)

for identified charged pions and kaons from Au+Au collisions atradic119904NN = 200GeV and pp collisions at radic119904NN = 200GeV where 120590 isthe width Filtered HIJING and UrQMD calculations are shown forpions and kaons from Au+Au collisions at radic119904NN = 200GeV Valuesare shown forradic2119898119879kin from Au+Au collisions where119898 is the massof a pion or a kaon and 119879kin is calculated from identified particlespectra [41] The width predicted by the blast-wave model [39] isalso shown for pions The graph is taken from [28]

the data from ALICE [37] Figure 21 [28] compares betweenBF calculated from HIJING and blast-wave model Thedetailed HIJING results are discussed in [47]

48 PYTHIA The PYTHIA is designed to generate high-energy-physics ldquoeventsrdquo that is sets of outgoing particlesproduced in the interactions between two incoming particlesThe objective is to provide as accurate as possible a represen-tation of event properties in a wide range of reactions withinand beyond the Standard Model with emphasis on thosewhere strong interactions play a role directly or indirectly

and therefore multihadronic final states are produced [94]The PYTHIA 572 is an event generator one can study theproton-proton collision events that are generated at differentcenter of mass (cm) energies [45]This can be shown clearlyat different energies in Figure 22 [45] Then the width of BFcan be studied for different multiplicity bins

The results presented in [28 54] show that the stringfragmentation implemented in PYTHIA describes the pro-duction particles and their charge balance functions Theydeduced from measured 119861(120575120578120578

119908) at six different windows

119861(120575120578) for the six windows 120578119908coincides with each other It was

shown that the scaled balance functions is corresponding toBF in the whole pseudorapidity range 119861(Δ120578 | infin) [54]

49 AMPT Model A multiphase transport (AMPT) is aMonte Carlo transport model for heavy ion collisions atrelativistic energies written in FORTRAN 77 It uses HIJINGfor generating the initial conditions Zhangrsquos Parton Cas-cade (ZPC) for modelling the partonic scatterings and arelativistic transport (ART) model for treating hadronicscatterings The AMPT model consists of four parts [95]the initial conditions which are obtained from HIJINGpartonic interactions the transition from the partonic case tothe hadronic matter case and hadronic interactions AMPTmodel uses the coalescence model to coalesce partons tocreate hadrons

It was shown in [54] that BF do not depend on the size andposition of the windows and are consistent with the resultsof pp in PYTHIA The charge balance functions are boost-invariance in both hadron-hadron and nuclear interactionThe boost invariance can scale BF with the window sizewithin the whole range of the rapidity Therefore BF aregood measures free from the restriction of finite longitudinalacceptanceThe dependence on transverse momentum of thelongitudinal property of balance functions is a sensitive probefor charge balance in hadronization mechanism

5 Discussion and Conclusions

The main topics of this review are the study of correla-tions between opposite-sign charge pairs Together with theparticle-ratio fluctuations these can provide a powerful toolto probe dynamics and properties of QGP beside hadroniza-tion and particle production It has been suggested that theexistence of a QCD phase transition would cause an increaseand divergence of fluctuations Thus the fluctuations couldbe used to study various particlecharge fluctuations near theQCD critical end point (CEP) On the other hand BF whichmeasure the correlations between opposite-sign charge pairsis sensitive to the mechanisms of charge formation and thesubsequent relative diffusion of the balancing charges Theirstudy can provide information about charge creation time aswell as the subsequent collective behavior of particles

In this review we have attempted to explain most ofthe important aspects of BF in high-energy physics Thevarious definitions are introduced and confronted to differentexperimental measurements and the effective models Theessential points we focused on is BF including the advantagesand short-comings Then we have discussed the various

20 Advances in High Energy Physics⟨Δ

120578⟩

Centrality percentile0 20 40 60 80


08

06

04

ALICEHIJINGAMPT (string melting)

AMPT (string melting wo rescattering)AMPT (default)

(a)

Centrality percentile

⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)

Figure 20 The centrality dependence of the widths of BF ⟨Δ120578⟩ and ⟨Δ120601⟩ for the correlations studied in terms of the relative pseudorapidityand the relative azimuthal angle respectively The data points are compared to the predictions from HIJING [42 43] and AMPT [44] Thegraph is taken from [37]

Blast wave modelHIJING filteredData 0ndash5

B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2

Figure 21 The balance functions in terms of Δ119910 for identifiedcharged pion pairs from Au+Au collisions at radic119904NN = 200GeV fornine centrality bins The graph is taken from [28]

experimental measurements depending on different quanti-ties for example the system size centrality and the beamenergyThe theoretical models describing and calculating BFhave been discussed

nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882

Figure 22 The width of balance functions in the rapidity region[minus3 3] for different multiplicities in pp collision at radic119904NN = 22 64130 200GeV The graph is taken from [45]

Three main results can be extracted from this reviewFirst BF have been calculated in terms of rapidity windowsize and pseudorapidity as given in Figure 3 Second BF interms of the reaction centrality and the beam energy (center-of-mass energy) are shown in Figures 5 and 6 Third BF


in terms of the invariant momentum 119902inv are also studiedBF were measured in various system sizes for exampleAu+Au at radic119904NN = 200GeV in the STAR experiment [54]and Pb+Pb collisions at radic119904NN = 172GeV in the ALICEexperiment [37] Also the width of BF was measured inPb+Pb C+C and Si+Si collisions atradic119904NN = 88 and 172 GeVat SPS [16] The calculations from different effective modelshave been calculated and compared with the data Figures12 13 and 17 Recent results depending on the system sizeand centrality for all charged particles have been studied atradic119904NN = 173GeV for p-p C-C Si-Si and Pb-Pb collisions[26 27 35] The dependence on the rapidity and the beamenergies are also studied [35 96]WhileHIJINGandUrQMDmodels fail to reproduce the narrowing in the balancefunction width observed [35] AMPT does The net-chargefluctuations are studied at LHC [97] for event-by-event net-charge fluctuations in terms of the pseudorapidity Δ120578 andazimuthal angle Δ120601 in Pb-Pb collisions at radic119904NN = 276TeVThe balance functions confronted to the STAR results showthat the quark chemistry can be determined The resultsagree within 20 with the expectations [33] This providesquantitative highlights on the chemical evolution of the QGPfor example the femtoscopy of two-particle correlationsThisstudy should be extended with new experiment results fromSTAR ALICE CMS and ATLAS [33]

The main conclusions can be summarized as follows

(i) the effective models are well suited to calculate thebalance functions

(ii) the most important quantities are the rapidity andpseudorapidity

(iii) the balance functions are very sensitive to the interac-tion centrality but not for the beam energy

(iv) the balance function width seems to be related to thehadronization time

(v) the balance functions can estimate the hadronizationtime from the jets production in p+p collision

(vi) the phase transition from hadron to quark matterand the properties of such matter the correlationsbetween charge and anticharge can be studieddirectly

Conflict of Interests

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

References

[1] M Gyulassy ldquoThe QGP discovered at RHICrdquo in Proceed-ings of the NATO Advanced Study Institute Structure andDynamics of Elementary Matter Kemer Turkey October 2003httparxivorgabsnucl-th0403032

[2] KAdcoxbd S S Adlere S Afanasiev et al ldquoFormation of densepartonic matter in relativistic nucleus-nucleus collisions atRHIC experimental evaluation by the PHENIXCollaborationrdquoNuclear Physics A vol 757 no 1-2 pp 184ndash283 2005

[3] B B Back M D Baker M Ballintijn et al ldquoThe PHOBOSperspective on discoveries at RHICrdquoNuclear Physics A vol 757no 1-2 pp 28ndash101 2005

[4] J Adams M M Aggarwala Z Ahammed et al ldquoExperimentaland theoretical challenges in the search for the quarkndashgluonplasma the STAR Collaborationrsquos critical assessment of theevidence from RHIC collisionsrdquo Nuclear Physics A vol 757 pp102ndash183 2005

[5] I Arsene I G Bearden D Beavis et al ldquoQuark-gluon plasmaand color glass condensate at RHIC The perspective from theBRAHMS experimentrdquo Nuclear Physics A vol 757 no 1-2 pp1ndash27 2005

[6] D Blaschke andC Pena ldquoQuarkonia andQGP studiesrdquoNuclearPhysics BmdashProceedings Supplements vol 214 no 1 pp 137ndash1422011

[7] T Matsui and H Satz ldquoJ120595 suppression by quark-gluon plasmaformationrdquo Physics Letters B vol 178 no 4 pp 416ndash422 1986

[8] R Rapp D Blaschke and P Crochet ldquoCharmonium andbottomonium in heavy-ion collisionsrdquo Progress in Particle andNuclear Physics vol 65 no 2 pp 209ndash266 2010

[9] N Brambilla S Eidelman B K Heltsley et al ldquoHeavy quarko-nium progress puzzles and opportunitiesrdquo The EuropeanPhysical Journal C vol 71 article 1534 2011

[10] M Bedjidian D Blaschke G T Bodwin et al ldquoHard probesin heavy ion collisions at the LHC heavy flavour physicsrdquohttparxivorgabshep-ph0311048

[11] S A Voloshin ldquoAnisotropic flow at RHIC constituent quarkscalingrdquo Journal of Physics Conference Series vol 9 no 1 article276 2005

[12] C-YWong ldquoSignatures of quark-gluonplasmaphase transitionin high-energy nuclear collisionsrdquo Nuclear Physics A vol 681no 1ndash4 pp 22ndash33 2001

[13] M I Gorenstein ldquoQuark-gluon plasma signatures in nucleus-nucleus collisions at CERN SPSrdquoActa Physica Hungarica SeriesA Heavy Ion Physics vol 14 no 1ndash4 pp 141ndash148 2001

[14] I Arsenej I G Bearden D Beavis et al ldquoResults from the first3 years at RHICmanaged for the US Department of Energy byBrookhaven Science Associates a company founded by StonyBrook University and Battellerdquo 2005

[15] S A Bass P Danielewicz and S Pratt ldquoClocking hadronizationin relativistic heavy-ion collisions with balance functionsrdquoPhysical Review Letters vol 85 no 13 pp 2689ndash2692 2000

[16] P Christakoglou A Petridis and M Vassiliou ldquoEnergyand rapidity dependence of electric charge correlationsat 20ndash158GeV beam energies at the CERN SPS(NA49)rdquo Published in the Proceedings of the ISMD05httparxivorgabsnucl-ex0510045

[17] D Drijard H G Fischer W Geist et al ldquoQuantum numbereffects in events with a charged particle of large transversemomentum (II) Charge correlations in jetsrdquo Nuclear PhysicsB vol 166 no 2 pp 233ndash242 1980

[18] D Drijard H G Fischer R Gokieli et al ldquoDensity chargeand transverse momentum correlations of particles in non-diffractive proton-proton collissions atradic119904 = 525GeVrdquoNuclearPhysics B vol 155 no 2 pp 269ndash294 1979

[19] I V Ajinenko S G Baladyan Y A Belokopytov et al ldquoChargeand energy flow in 120587+p K+p and pp interactions at 250GeVcrdquoZeitschrift fur Physik C Particles and Fields vol 43 pp 37ndash441989

[20] R Brandelik W Braunschweig K Gather et al ldquoEvidence forcharged primary partons in 119890+119890minus rarr 2 jetsrdquo Physics Letters Bvol 100 pp 357ndash363 1981


[21] M Althoff R Brandelik W Braunschweig et al ldquoChargedhadron composition of the final state in e+eminus annihilation athigh-energiesrdquo Zeitschrift fur Physik C Particles and Fields vol17 no 1 pp 5ndash15 1983

[22] H Aihara M Alston-Garnjost D H Badtke et al ldquoObserva-tion of strangeness correlations in e+eminus annihilation atradic119904 = 29GeVrdquo Physical Review Letters vol 53 no 23 pp 2199ndash22021984

[23] H Aihara M Alston-Garnjost R E Avery et al ldquoStudy ofbaryon correlations in e+eminus annihilation at 29GeVrdquo PhysicalReview Letters vol 57 p 3140 1986

[24] P D Acton G Alexander J Allison et al ldquoEvidence for chain-like production of strange baryon pairs in jetsrdquo Physics LettersB vol 305 pp 415ndash427 1993

[25] J Adams C Adler Z Ahammed et al ldquoNarrowing of thebalance function with centrality in Au+Au collisions atradic119904119873119873

=

130 GeVrdquo Physical Review Letters vol 90 Article ID 1723012003

[26] C Alt T Anticic B Baatar et al ldquoSystem size and centralitydependence of the balance function in 119860 + 119860 collisions atradic119904NN = 172GeVrdquo Physical Review C vol 71 Article ID 0349032005

[27] P Christakoglou A Petridis andM Vassiliou ldquoSystem size andcentrality dependence of the electric charge correlations inA+Aand p+p collisions at the SPS energiesrdquo Nuclear Physics A vol749 pp 279ndash282 2005

[28] H Wang Study of particle ratio fluctuations and charge balancefunctions at RHIC [PhD thesis] Michigan State University2013 httparxivorgabs13042073

[29] S Pratt ldquoAlternative contributions to the angular correla-tions observed at RHIC associated with parity fluctuationsrdquohttparxivorgabs10021758

[30] J Song F-L Shao and Z-T Liang ldquoQuark charge balancefunction and hadronization effects in relativistic heavy ioncollisionsrdquo Physical Review C vol 86 no 6 Article ID 0649039 pages 2012

[31] S Borsanyi Z Fodor S D Katz S Krieg C Ratti and K SzaboldquoFluctuations of conserved charges at finite temperature fromlattice QCDrdquo Journal of High Energy Physics vol 2012 no 1article 138 2012

[32] C Ratti R Bellwied M Cristoforetti and M Barbaro ldquoArethere hadronic bound states above the QCD transition temper-aturerdquo Physical Review D vol 85 no 1 Article ID 014004 8pages 2012

[33] S Pratt ldquoViewing the chemical evolution of the Quark-Gluonplasma with charge balance functionsrdquo Proceedings of lsquoCriticalPoint and the Onset of Deconfinementrsquo Napa Calif USAMarch 2013

[34] H Wang ldquoReaction plane and beam energy dependence of thebalance function at RHICrdquo Journal of Physics Conference Seriesvol 316 Article ID 012021 2011

[35] M M Aggarwal Z Ahammed A V Alakhverdyants et alldquoBalance functions from 119860119906 + 119860119906 119889 + 119860119906 and 119901 + 119901 collisionsat radic119904119873119873 = 200GeVrdquo Physical Review C vol 82 no 2 ArticleID 024905 16 pages 2010

[36] S Jeon and S Pratt ldquoBalance functions correlations chargefluctuations and interferometryrdquo Physical Review C vol 65 no4 Article ID 044902 6 pages 2002

[37] B Abelev J Adam D Adamova et al ldquoCharge correlationsusing the balance function in Pb-Pb collisions at radic119904119873119873

= 276TeVrdquo Physics Letters B vol 723 no 4-5 pp 267ndash279 2013

[38] P BozekW Broniowski andW Florkowski ldquoBalance functionsin a thermal model with resonancesrdquo Acta Physica HungaricaSeries A Heavy Ion Physics vol 22 no 1-2 pp 149ndash157 2005

[39] S Cheng S Petriconi S Pratt et al ldquoStatistical and dynamicmodels of charge balance functionsrdquo Physical Review CmdashNuclear Physics vol 69 no 5 Article ID 054906 2004

[40] F Retiere andM A Lisa ldquoObservable implications of geometri-cal and dynamical aspects of freeze-out in heavy ion collisionsrdquoPhysical Review C vol 70 no 4 Article ID 044907 33 pages2004

[41] J Adams C Adler and M M Aggarwal ldquoIdentified particledistributions in 119901119901 and119860119906+119860119906 collisions atradic119904119873119873

= 200GeVrdquoPhysical Review Letters vol 92 no 11 Article ID 112301 6 pages2004

[42] X-N Wang and M Gyulassy ldquoHIJING 10 a Monte Carloprogram for parton and particle production in high energyhadronic and nuclear collisionsrdquo Computer Physics Communi-cations vol 83 no 2-3 pp 307ndash331 1994

[43] X-NWang andM Gyulassy ldquoHijing a Monte Carlo model formultiple jet production in pp pA and AA collisionsrdquo PhysicalReview D vol 44 no 11 pp 3501ndash3516 1991

[44] B Zhang C M Ko B-A Li and Z Lin ldquoMultiphase transportmodel for relativistic nuclear collisionsrdquo Physical Review C vol61 Article ID 067901 2000

[45] J Du N Li and L Liu ldquoNarrowing of the charge balance func-tion and hadronization time in relativistic heavy-ion collisionsrdquoPhysical Review C vol 75 Article ID 021903 2007

[46] J Letessier and J Rafelski Hadron and Quark-Gluon PlasmaCambridge University Press Cambridge UK 2004

[47] S Cheng Modelling relativistic heavy-ion collisions [PhD the-sis] Michigan State University 2002

[48] D Drijard H G Fischer W Geist et al ldquoQuantum numbereffects in events with a charged particle of large transversemomentum (I) Leading particles in single and diquark jetsrdquoNuclear Physics B vol 156 no 2 pp 309ndash327 1979

[49] T A Trainor ldquoWhat does the balance function measurerdquohttparxivorgabshep-ph0301122

[50] A Bialas and J Rafelski ldquoBalance of baryon number in thequark coalescence modelrdquo Physics Letters B vol 633 no 4-5pp 488ndash491 2006

[51] J E Morel and J D Edwards ldquoThe comoving-frame andlaboratory-frame nonequilibrium grey radiation diffusionapproximations in the nonrelativistic limitrdquo in Proceedings ofthe Conference on Numerical Methods for Multimaterial FluidFlows Prague Czech Republic September 2007

[52] B I Abelev M M Aggarwal Z Ahammed et al ldquoAzimuthalcharged-particle correlations and possible local strong parityviolationrdquo Physical Review Letters vol 103 Article ID 2516012009

[53] B I Abelev M M Aggarwal Z Ahammed et al ldquoObservationof charge-dependent azimuthal correlations and possible localstrong parity violation in heavy ion collisionsrdquo Physical ReviewC vol 81 Article ID 054908 2010

[54] N Li Azimuthal anisotropy and longitudinal property of chargebalance function in relativistic heavy ion collisions [PhD thesis]Huazhong Normal University Wuhan China 2010

[55] S Pratt ldquoGeneral charge balance functions a tool for studyingthe chemical evolution of the quark-gluon plasmardquo PhysicalReview C vol 85 no 1 Article ID 014904 11 pages 2012

[56] M R Atayan Y Bai E A de Wolf et al ldquoBoost invariance andmultiplicity dependence of the charge balance function in 120587+p


and K+p collisions at radic119904 = 22 GeVrdquo Physics Letters B vol 637no 1-2 pp 39ndash42 2006

[57] L Zhiming L Na L Lianshou and W Yuanfang ldquoPseudora-pidity and transversemomentumdependence of charge balancein Au-Au collisions at radic119904119873119873

= 200 GeVrdquo International Journalof Modern Physics E vol 16 no 10 pp 3347ndash3354 2007

[58] N Li Z Li and Y Wu ldquoLongitudinal boost invariance of thecharge balance function in hadron-hadron and nucleus-nucleuscollisionsrdquo Physical Review C vol 80 Article ID 064910 2009

[59] S Jeon and V Koch ldquoEvent-by-event fluctuationsrdquo Review forlsquoQuark-Gluon Plasma 3rsquo eds RC Hwa and X-NWangWorldScientific Singapore httparxivorgabshep-ph0304012

[60] H Tydesj Net-charge fluctuations in ultra-relativitic nucleus-nucleus collisions (Licentiate thesis) Lund University 2003

[61] P Levai T S Biro T Csorgo and J Zimanyi ldquoSimple pre-dictions from ALCOR

119888for rehadronization of charmed quark

matterrdquo New Journal of Physics vol 2 article 32 2000[62] M C Abreau B Alessandro and C Alex ldquo119869120595 and Drell-

Yan cross-sections in Pb-Pb interactions at 158GeVcrdquo PhysicsLetters B vol 410 no 2ndash4 pp 327ndash336 1997

[63] A Bialas ldquoBalance functions in coalescence modelrdquo PhysicsLetters B vol 579 no 1-2 pp 31ndash38 2004

[64] T S Biro P Levai and J Zimanyi ldquoALCOR a dynamical modelfor hadronizationrdquo Physics Letters B vol 347 no 1-2 pp 6ndash121995

[65] J Zimanyi T S Biro T Csorgo and P Levai ldquoParticle spectrafrom the ALCOR modelrdquo Acta Physica Hungarica New SeriesHeavy Ion Physics vol 4 no 1ndash4 pp 15ndash32 1996

[66] T S Biro P Levai and J Zimanyi ldquoStrange hadrons from theALCOR rehadronizationmodelrdquo inProceedings of theWorkshopon Strangeness in Hadronic Matter (Strangeness rsquo95) vol 340 ofAIP Conference Proceedings p 405 Tucson Ariz USA 1995

[67] A Bialas ldquoQuark model and strange baryon production inheavy ion collisionsrdquoPhysics Letters B vol 442 no 1ndash4 pp 449ndash452 1998

[68] J Zimanyi T S Biro T Csorgo and P Levai ldquoQuark liberationand coalescence at CERN SPSrdquo Physics Letters B vol 472 no3-4 pp 243ndash246 2000

[69] S Schlichting and S Pratt ldquoCharge conservation at energiesavailable at the BNL relativistic heavy ion collider and contri-butions to local parity violation observablesrdquo Physical ReviewC vol 83 Article ID 014913 2011

[70] A N Tawfik ldquoEquilibrium statistical-thermal models in high-energy physicsrdquo International Journal of Modern Physics A vol29 no 17 Article ID 1430021 2014

[71] F Karsch K Redlich and A Tawfik ldquoHadron resonance massspectrum and lattice QCD thermodynamicsrdquo The EuropeanPhysical Journal C vol 29 pp 549ndash556 2003

[72] F Karsch K Redlich and A Tawfik ldquoThermodynamics atnon-zero Baryon number density a comparison of lattice andHadron resonance gas model calculationsrdquo Physics Letters Bvol 571 pp 67ndash74 2003

[73] K Redlich F Karsch and A Tawfik ldquoHeavy-ion collisionsand lattice QCD at finite baryon densityrdquo Journal of PhysicsG Nuclear and Particle Physics vol 30 no 8 pp S1271ndashS12742004

[74] A Tawfik ldquoQCD phase diagram a comparison of lattice andhadron resonance gas model calculationsrdquo Physical Review Dvol 71 Article ID 054502 2005

[75] A Tawfik ldquoInfluence of strange quarks on the QCD phasediagram and chemical freeze-outrdquo Journal of Physics G Nuclearand Particle Physics vol 31 no 6 pp S1105ndashS1110 2005

[76] H Sorge H Stocker and W Greiner ldquoPoincare invariantHamiltonian dynamics modelling multi-hadronic interactionsin a phase space approachrdquo Annals of Physics vol 192 pp 266ndash306 1989

[77] C Adler Z Ahammed C Allgower et al ldquoIdentified particleelliptic flow in 119860119906 + 119860119906 collisions atradic119904119873119873

= 130 GeVrdquo PhysicalReview Letters vol 87 Article ID 182301 2001

[78] P J Siemens and J O Rasmussen ldquoEvidence for a blast wavefrom compressed nuclear matterrdquo Physical Review Letters vol42 no 14 pp 880ndash883 1979

[79] R Snellings ldquoElliptic flow a brief reviewrdquo New Journal ofPhysics vol 13 Article ID 055008 2011

[80] M Miller and R Snellings ldquoEccentricity fluctuationsand its possible effect on elliptic flow measurementsrdquohttparxivorgabsnucl-ex0312008

[81] P Sorensen ldquoElliptic flow a study of space-momentum corre-lations in relativistic nuclear collisionsrdquo review article writtenfor the QGP4 book edited by Rudy Hwa and Xin-Nian Wanghttparxivorgabs09050174

[82] R A Lacey R Wei J Jia N N Ajitanand J M Alexanderand A Taranenko ldquoInitial eccentricity fluctuations and theirrelation to higher-order flowharmonicsrdquoPhysical ReviewC vol83 Article ID 044902 2011

[83] F Retiere ldquoTwo-particle correlations in radic119904119873119873= 130GeVrdquo in

Proceedings of the International Workshop on the Physics of theQuark-Gluon Plasma Palaiseau France September 2001

[84] Y M Sinyukov S V Akkelin and N Xu ldquoFinal conditions inhigh energy heavy ion collisionsrdquo Physical Review CmdashNuclearPhysics vol 59 no 6 pp 3437ndash3440 1999

[85] E Schnedermann J Sofffrank and U Heinz ldquoThermal phe-nomenology of hadrons from 200A GeV S+S collisionsrdquo Physi-cal Review C vol 48 p 2462 1993

[86] R Hanbury Brown and R Q Twiss ldquoLXXIV A new typeof interferometer for use in radio astronomyrdquo PhilosophicalMagazine vol 45 no 366 pp 633ndash682 1954

[87] R H Brown and R Q Twiss ldquoCorrelation between photons intwo coherent beams of lightrdquo Nature vol 177 pp 27ndash29 1956

[88] R Hanbury Brown and R Q Twiss ldquoA test of a new type ofstellar interferometer on Siriusrdquo Nature vol 178 no 4541 pp1046ndash1048 1956

[89] S S Padula ldquoHBT interferometry historical perspectiverdquoBrazilian Journal of Physics vol 35 no 1 pp 70ndash99 2005

[90] U Heinz ldquoHanbury Brown-Twiss interferometry in highenergy nuclear and particle physicsrdquo Overview talkgiven at CRISrsquo98 (Catania June 8ndash12 1998) Singaporehttparxivorgabshep-ph9806512

[91] B I Abelev MM Aggarwal Z Ahammed et al ldquoLongitudinalscaling property of the charge balance function in Au+Aucollisions atView theMathML sourceradic119904119873119873

= 200GeVrdquoPhysicsLetters B vol 690 pp 239ndash244 2010

[92] A Bialas ldquoBalance functions reexaminedrdquo Physical Review Cvol 83 Article ID 024914 2011

[93] X-N Wang ldquoA pQCD-based approach to parton productionand equilibration in high-energy nuclear collisionsrdquo PhysicsReports vol 280 no 5-6 pp 287ndash371 1997

[94] T Sjostrand S Mrenna and P Skands ldquoPYTHIA 64 physicsand manualrdquo Journal of High Energy Physics vol 2006 no 5article 026 2006

[95] Z-W Lin C M Ko B-A Li B Zhang and S Pal ldquoMultiphasetransport model for relativistic heavy ion collisionsrdquo PhysicalReview C vol 72 Article ID 064901 2005


[96] C Alt T Anticic B Baatar et al ldquoRapidity and energydependence of the electric charge correlations inA+A collisionsfrom 20A to 158A GeVrdquo Physical Review C vol 76 Article ID02914 2007

[97] MWeber ldquoNet-charge fluctuations and balance functions at theLHCrdquo Nuclear Physics A vol 904 pp 467cndash470c 2013

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


t

120591f

Hadron gas

QGP

BeamBeam

1205910

z

QCD phasetransition

ThermalizationParton cascade

Kinetic freeze-out

Chemical freeze-out

Figure 1The space-time evolution of heavy-ion collisionThefigureis taken from [28]

(ii) Lattice QCD results predicted two orders for phasetransition(s) It is argued that a first-order phasetransition is likely in system consisting of two flavorswhile a second-order one is likely in the three-flavorsystem Furthermore a smooth cross-over was seenin the QCD simulations Linking such theoreticalpredictions with the experimental results would bepossible through varying the critical temperature Forinstance at low temperature the matter is confinedthat is hadronic phase while at high temperatureQGP phase is likely [12]

(iii) The strangeness enhancement at alternating gradientsynchrotron (AGS) is found larger than that at superproton synchrotron (SPS) which obviously seems toweaken the concept of strangeness enhancement asa signal of QGP [13] Nevertheless the search forenhancement at RHIC and LHC energies should becontinued

(iv) The estimation of the time span till equilibrationrefers to very small value (sim10 fmc) Thus the evo-lution of the equilibrated states cannot be evident[14] Thus it would not be possible to assure thatthe hadronic phase was originated in a partonic state(prior to hadronization) [14] The situation becomesmore drastic at RHIC and LHC energies The criticaland freeze-out temperatures become almost indistin-guishable [14]

The balance functions (BF) were proposed by Bass et al[15] as a measure for the correlation of the positive and nega-tive charged particles produced during the relativistic heavy-ion collisionsTheir width can be related to the hadronizationtime The charge correlation functions which are devoted tostudy the jets hadronization [16] are used to derive BF Sofar they have been estimated in pp collisions at intersectingstorage rings (ISR) [17ndash19] e+ + eminus annihilation at PETRA atDESY [20ndash24] Au+Au in STAR experiment at BNL RHIC[25] and Pb+Pb in NA49 experiment at CERN SPS [26 27]Due to charge conservation oppositely charged particles areproduced in pairs But the produced pairs are separated in therapidity region due to their different momenta This implies

that BF can be extracted from the fact that the pairs ofopposite charges are created in the local space This ideadefines how to proceed with the measurement of balancebetween produced pairs

The different heavy-ion experiments can be differentiatedaccording to the collision energy or nucleon-nucleon (NN)center-of-mass energyradic119904NN [46] the system size and type ofreactants whether being elementary NN or nucleus-nucleus(AA) collisions

119910 =1

2ln(

119864 + 119901119871

119864 minus 119901119871

) = ln(119864 + 119901

119871

119898perp

) (1)

where119901119871is the longitudinalmomentumand119898

perp= radic1198982 + 1199012

119879

is the transverse mass The Lorentz boosts are the trans-formations with respect to one of three dimensions takingas the frame of reference At ultrarelativistic energies itis convenient to deal with the pseudorapidity 120578 which isdefined in analogy to 119910 (1)

120578 = minus ln [tan(1205792)] (2)

where 120579 is the angle of emitted particles relative to the beamaxis

The present work is organized as follows Section 1presents a general overview about the history of QGPSection 2 is devoted to the various definitions of BF Theexperimental measurements will be discussed in Section 3Section 4 discusses some effective models used to calculateBF in high-energy physics Finally Section 5 presents thediscussion and conclusions

2 Definitions

In relativistic heavy-ion collisions it is assumed that manyproduced particles of different charges expand in temporaland spatial dimensions [39] Due to charge conservationboth positive and negative charges have to be produced inthe same space-time during the evolution of the mediumThe correlation between the opposite charges is characterizedthrough BF which apparently measure the balance betweenboth types of charges [47] In early studies Bass et al [15] haveproposed that BF are signatures differentiating between early-and late-stage of the hadronization The balance functionsare proposed to work as a ldquoclockrdquo determining whether thequark production occurred at early times 119905 lt 1 fmc orat late-stage [15] For charges created in the early stagebalancing charges are separated by the order of one unit ofrapidity while those formed in a late stage are far from thecorrelation Delayed hadronizationmeans that theQGP staysfor a long time This implies that the QGP might be formedat a certain time before the evolution of the hot matter Inprincipal BF were proposed to investigate the hadronizationfrom jets production in proton-proton collisions [17 18] Ina series of papers [17 18 48] BF were associated with chargecorrelations

Furthermore the conditional probability is the probabil-ity that an event will occur under some conditions while



119886should be


119887minus 119910

119886



119861 (1198751| 119875

2) equiv1

2120588 (119887 119875

2| 119886 119875

1) minus 120588 (119887 119875

2| 119887 119875

1)

+ 120588 (119886 1198752| 119887 119875

1) minus 120588 (119886 119875

2| 119886 119875

1)

(3)

where 120588(119887 1198752| 119886 119875





119861 (1198751| 119875

2) equiv1

2

119873minus+(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

+minus(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

2)

(4)


1 119875



+minus119873

minusminusand119873

++


119861 (Δ2| Δ

1) =119863 (+ Δ

2| minus Δ

1) minus 119863 (+ Δ

2| + Δ

1)

119873+(Δ

1)

+119863 (minus Δ

2| + Δ


2| minus Δ

1)

119873minus(Δ

1)

(5)


2| minus Δ

1)119873

+(Δ

1) or 120588(119887 Δ

2| 119886 Δ

1) =

119873(119887 Δ2| 119886 Δ

1)119873(119886 Δ




119873119894(Δ) = int

Δ

119889119899119894

119889119910119889119910

119863 (119894 Δ2| 119895 Δ

1) = int

Δ2

1198891199102intΔ1

1198891199101

1198892119899119894119895

1198891199101119889119910

2

(6)



2le Δ and


2equiv Δ (119910

1+ 119910



120588 (119887 Δ2| 119886 Δ

1) =119873 (119887 Δ

2| 119886 Δ

1)

119873 (119886 Δ1)

(7)

where

119873(119887 Δ2| 119886 Δ

1) = int

Δ1

1198893

1199011intΔ2

1198893

1199012119891119886119887(119901

1 119901

2)

119873 (119886 Δ1) = int

Δ1

1198893

1199011119891119886(119901

1)

(8)

where 119891119886(119901

1) or 119891

119887(119901



1 119901

2) is the two-



1 119901

2) can


11989111990211199022

(1199011 119901

2) = 119873

1199021

1198731199022

1198991199021

(1199011) 119899

1199022

(1199012) (1 minus 120575

11990211199022

)

+ 1198731199021

(1198731199022

minus 1) 1198991199022

(1199012) 120575

11990211199022

11989111990211199022

(1199011 119901

2) = 119873

1199021

1198731199022

1198991199021

(1199011) 119899

1199022

(1199012) (1 minus 120575

11990211199022

)

+ 1198731199021

(1198731199022

minus 1) 1198991199022

(1199012) 120575

11990211199022

(9)

where 1199021and 119902



11989111990211199022

(1199011 119901

2)

= 1198731199021

1198731199022

1198991199021

(1199011) 119899

1199022

(1199012)

+ 1198731199021

[119899pair119902119902(119901

1 119901

2) minus 119899

1199021

(1199011) 119899

1199022

(1199012)] 120575

11990211199022

(10)


119891119865119861(119864 120573 120582 120574) =

1




119902119902

119899119902(119901

2) = int119889

3

1199011119899pair119902119902(119901

1 119901

2) (12)





0 Thus the term


0


119899lowast

119902(119901

lowast

) = 119899th (119901lowast

) =119890minus119864lowast119879

412058711989821198791198702(119898119879)

(13)





119872(Δ119910120575119910)

sum

119894=1

119899119886119894(120575119910) equiv 119873

119886(Δ119910) (14)


119899119886119894(120575119910) asymp int

119910119894+1205751199102

119910119894minus1205751199102

1205881119899119886(119910) 119889119910 (15)




defined as [49]

119861119870(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus 1198861198871

sum119872

119894=1119899119886119894

119872(Δ119910120575119910)minus119896

sum

119894=1

119899119886119894sdot (119899

119887(119894+119896)minus 120575

1198861198871205751198960)

119861119870(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus 1198861198871

sum119872

119894=1119899119886119894

119872(Δ119910120575119910)

sum

119894=1minus119896

119899119886119894sdot (119899

119887(119894+119896)minus 120575

1198861198871205751198960)

(16)



119875119894(119886) =

119899119886119894(119899

119887minus 120575

119886119887)

119873119886(119873

119887minus 120575

119886119887) 119875

119894119895(119886119887) =

119899119886119894(119899

119887119895minus 120575

119886119887120575119894119895)

119873119886(119873

119887minus 120575

119886119887)

(17)


1198751198951(119886119887 Δ119910 120575119910) equiv

119875119894119895(119886119887)

119875119894(119886)

=119899119886119894(119899

119887119895minus 120575

119886119887120575119894119895)

119899119886119894(119873

119887minus 120575

119886119887) (18)


119886119894(119899

119887119895minus

120575119886119887120575119894119895)


119875119896Δ119910(119886119887) equiv

119873119896(119886119887 Δ119910 120575119910)

119873119886(Δ119910) (119873

119887(Δ119910) minus 120575

119886119887)

119861119896(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus119886119887119875

119896Δ119910(119886119887)119873

119886(Δ119910) (119873

119887(Δ119910) minus 120575

119886119887)

119873119886(Δ119910)

119861119896(Δ119910 120575119910)

equiv1

2sum

119886

119875119896Δ119910(119886119886) minus

sum119886119887119875119896Δ119910(119886119887)119873

119886(Δ119910)119873

119887(Δ119910)

119873119886(Δ119910)

(19)



times10minus3

1

05

0

minus05

minus1

Most central ()70 60 50 40 30 20 10 0



⟨cos

(120601120572+120601120573minus2Ψ

RP)⟩



120574+minus= 119865

119876((sum

119894

cos 2120601119894⟨cos (Δ120601balance)⟩ (120601119894)



)

(20)


119894119901119894

119909= 0 sum

119894119901119894

119910= 0



119894

119905cos(120601


be written as [29]

120574 = minus119865119901

sum119894(cos2120601

119894minus sin2120601

119894)

1198722

tot (21)


119872++ 119872

minus+ 119872





0 05 10

01

02

03

04

05

06

07

minus05 lt y lt 05

0 lt y lt 1

1 lt y lt 2

15 lt y lt 25

B(Δ

y|yw)

Δy

(a)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

B(Δ

y|yw)

Δy

(b)

Bs(Δy)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

Δy

(c)











119861 (Δ119910 | 119910119908) = 119861 (Δ119910 | infin) (1 minus Δ119910) (22)






Pions

0

02

04

06 K0s

1205880

0 04 08 12 16

B(q

inv)

((G

eVc

)minus1)

qinv (GeVc)

(a)

Kaons02

01

0

0 04 08 12 16

120593

qinv (GeVc)

B(q

inv)

((G

eVc

)minus1)

(b)





⟨(1205751198762

)⟩ = ⟨1198762

⟩ minus ⟨119876⟩2

= 1199022

(⟨1198732

⟩ minus ⟨119873⟩2

) (23)

where119876 = 119899+minus119899



⟨119873ch⟩ ⟨1205751198772

⟩ = 4⟨(120575119876

2)⟩

⟨119873ch⟩ (24)

where

119877 =⟨119873

+119873


minus⟩ ⟨119873

+⟩

⟨119873minus⟩ ⟨119873

+⟩

(25)


119863 (119876) = 4⟨(120575119876)

2

⟩

119873ch (26)


119863 (119876)

4= 1 minus int

119910119908

0

119861 (Δ119910 | 119910119908) 119889Δ119910 +

⟨119876⟩

119873ch (27)



119861 (119902inv) = 1198861199022

inv119890minus1199022

inv21205902

(28)


++119896





77GeV 196GeV

27GeV 39GeV 624GeV

115GeV

B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06

DataShuffled

200GeV

Δ120578

B(Δ

120578)

0

02

04

06

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18







DataShuffled

DataShuffled

DataShuffled

B(Δ

120601)

Δ120601





0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

0 1 2Δ120601

0 1 2Δ120601

0 1 2 3

3

3

3

3

3

3

3

3Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2

Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2



120594119886119887=⟨119876

119886119876119887⟩

119881 (29)



120594QGP119886119887

= Δ119886119887(119899

119886+ 119899

119886) (30)

where 119899119886 119899



120594HG119886119887= sum

120572

119899120572119902120572119886119902120572119887 (31)

where 119902120572119886



119866120572120573(120578) = 4sum

119886119887119888119889

⟨119899120572⟩ 119902

120572119886120594(had)(minus1)119886119887

(0) 119892(had)119887119888

sdot (120578) 120594(had)(minus1)119888119889

(120578) 119902120573119889⟨119899

120573⟩

(32)


119861120572120573(Δ120578) =

119866120572120573(Δ120578)

119899120573+ 119899

120573

(33)





DataShuffled

DataShuffled

DataShuffled

Kaons

B(q

inv)

1205942ndf = 174838

120590 = 0501

1205942ndf = 807638

120590 = 0504

1205942ndf = 65438

120590 = 0518

1205942ndf = 251638

120590 = 0496

1205942ndf = 753538

120590 = 0509

1205942ndf = 680438

120590 = 0526

1205942ndf = 482238

120590 = 0503

1205942ndf = 963838

120590 = 0519

1205942ndf = 445938

120590 = 0530




03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1 2

2

2

2

2

2

2

2

2

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1



119861 (120601 Δ120601) =1

2


++120601 Δ120601

119873+(120601)

+Δ



119873minus(120601)

(34)





⟨Δ120578⟩ =sum

119894119861 (Δ120578

119894) Δ120578

119894

sum119894119861 (Δ120578

119894) (35)



0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs




+minus(Δ120578 | 120578max)

should be corrected


Δ120578

120578max) (36)





119902 in which


119872(119894 119895) and 119862



= 2119878ℎ+ 1 can be



119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895


119902119894

and 119887119902119895


119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)





119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)


119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)



77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119886should be


119887minus 119910

119886



119861 (1198751| 119875

2) equiv1

2120588 (119887 119875

2| 119886 119875

1) minus 120588 (119887 119875

2| 119887 119875

1)

+ 120588 (119886 1198752| 119887 119875

1) minus 120588 (119886 119875

2| 119886 119875

1)

(3)

where 120588(119887 1198752| 119886 119875





119861 (1198751| 119875

2) equiv1

2

119873minus+(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

+minus(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

2)

(4)


1 119875



+minus119873

minusminusand119873

++


119861 (Δ2| Δ

1) =119863 (+ Δ

2| minus Δ

1) minus 119863 (+ Δ

2| + Δ

1)

119873+(Δ

1)

+119863 (minus Δ

2| + Δ


2| minus Δ

1)

119873minus(Δ

1)

(5)


2| minus Δ

1)119873

+(Δ

1) or 120588(119887 Δ

2| 119886 Δ

1) =

119873(119887 Δ2| 119886 Δ

1)119873(119886 Δ




119873119894(Δ) = int

Δ

119889119899119894

119889119910119889119910

119863 (119894 Δ2| 119895 Δ

1) = int

Δ2

1198891199102intΔ1

1198891199101

1198892119899119894119895

1198891199101119889119910

2

(6)



2le Δ and


2equiv Δ (119910

1+ 119910



120588 (119887 Δ2| 119886 Δ

1) =119873 (119887 Δ

2| 119886 Δ

1)

119873 (119886 Δ1)

(7)

where

119873(119887 Δ2| 119886 Δ

1) = int

Δ1

1198893

1199011intΔ2

1198893

1199012119891119886119887(119901

1 119901

2)

119873 (119886 Δ1) = int

Δ1

1198893

1199011119891119886(119901

1)

(8)

where 119891119886(119901

1) or 119891

119887(119901



1 119901

2) is the two-



1 119901

2) can


11989111990211199022

(1199011 119901

2) = 119873

1199021

1198731199022

1198991199021

(1199011) 119899

1199022

(1199012) (1 minus 120575

11990211199022

)

+ 1198731199021

(1198731199022

minus 1) 1198991199022

(1199012) 120575

11990211199022

11989111990211199022

(1199011 119901

2) = 119873

1199021

1198731199022

1198991199021

(1199011) 119899

1199022

(1199012) (1 minus 120575

11990211199022

)

+ 1198731199021

(1198731199022

minus 1) 1198991199022

(1199012) 120575

11990211199022

(9)

where 1199021and 119902



11989111990211199022

(1199011 119901

2)

= 1198731199021

1198731199022

1198991199021

(1199011) 119899

1199022

(1199012)

+ 1198731199021

[119899pair119902119902(119901

1 119901

2) minus 119899

1199021

(1199011) 119899

1199022

(1199012)] 120575

11990211199022

(10)


119891119865119861(119864 120573 120582 120574) =

1




119902119902

119899119902(119901

2) = int119889

3

1199011119899pair119902119902(119901

1 119901

2) (12)





0 Thus the term


0


119899lowast

119902(119901

lowast

) = 119899th (119901lowast

) =119890minus119864lowast119879

412058711989821198791198702(119898119879)

(13)





119872(Δ119910120575119910)

sum

119894=1

119899119886119894(120575119910) equiv 119873

119886(Δ119910) (14)


119899119886119894(120575119910) asymp int

119910119894+1205751199102

119910119894minus1205751199102

1205881119899119886(119910) 119889119910 (15)




defined as [49]

119861119870(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus 1198861198871

sum119872

119894=1119899119886119894

119872(Δ119910120575119910)minus119896

sum

119894=1

119899119886119894sdot (119899

119887(119894+119896)minus 120575

1198861198871205751198960)

119861119870(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus 1198861198871

sum119872

119894=1119899119886119894

119872(Δ119910120575119910)

sum

119894=1minus119896

119899119886119894sdot (119899

119887(119894+119896)minus 120575

1198861198871205751198960)

(16)



119875119894(119886) =

119899119886119894(119899

119887minus 120575

119886119887)

119873119886(119873

119887minus 120575

119886119887) 119875

119894119895(119886119887) =

119899119886119894(119899

119887119895minus 120575

119886119887120575119894119895)

119873119886(119873

119887minus 120575

119886119887)

(17)


1198751198951(119886119887 Δ119910 120575119910) equiv

119875119894119895(119886119887)

119875119894(119886)

=119899119886119894(119899

119887119895minus 120575

119886119887120575119894119895)

119899119886119894(119873

119887minus 120575

119886119887) (18)


119886119894(119899

119887119895minus

120575119886119887120575119894119895)


119875119896Δ119910(119886119887) equiv

119873119896(119886119887 Δ119910 120575119910)

119873119886(Δ119910) (119873

119887(Δ119910) minus 120575

119886119887)

119861119896(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus119886119887119875

119896Δ119910(119886119887)119873

119886(Δ119910) (119873

119887(Δ119910) minus 120575

119886119887)

119873119886(Δ119910)

119861119896(Δ119910 120575119910)

equiv1

2sum

119886

119875119896Δ119910(119886119886) minus

sum119886119887119875119896Δ119910(119886119887)119873

119886(Δ119910)119873

119887(Δ119910)

119873119886(Δ119910)

(19)



times10minus3

1

05

0

minus05

minus1

Most central ()70 60 50 40 30 20 10 0



⟨cos

(120601120572+120601120573minus2Ψ

RP)⟩



120574+minus= 119865

119876((sum

119894

cos 2120601119894⟨cos (Δ120601balance)⟩ (120601119894)



)

(20)


119894119901119894

119909= 0 sum

119894119901119894

119910= 0



119894

119905cos(120601


be written as [29]

120574 = minus119865119901

sum119894(cos2120601

119894minus sin2120601

119894)

1198722

tot (21)


119872++ 119872

minus+ 119872





0 05 10

01

02

03

04

05

06

07

minus05 lt y lt 05

0 lt y lt 1

1 lt y lt 2

15 lt y lt 25

B(Δ

y|yw)

Δy

(a)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

B(Δ

y|yw)

Δy

(b)

Bs(Δy)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

Δy

(c)











119861 (Δ119910 | 119910119908) = 119861 (Δ119910 | infin) (1 minus Δ119910) (22)






Pions

0

02

04

06 K0s

1205880

0 04 08 12 16

B(q

inv)

((G

eVc

)minus1)

qinv (GeVc)

(a)

Kaons02

01

0

0 04 08 12 16

120593

qinv (GeVc)

B(q

inv)

((G

eVc

)minus1)

(b)





⟨(1205751198762

)⟩ = ⟨1198762

⟩ minus ⟨119876⟩2

= 1199022

(⟨1198732

⟩ minus ⟨119873⟩2

) (23)

where119876 = 119899+minus119899



⟨119873ch⟩ ⟨1205751198772

⟩ = 4⟨(120575119876

2)⟩

⟨119873ch⟩ (24)

where

119877 =⟨119873

+119873


minus⟩ ⟨119873

+⟩

⟨119873minus⟩ ⟨119873

+⟩

(25)


119863 (119876) = 4⟨(120575119876)

2

⟩

119873ch (26)


119863 (119876)

4= 1 minus int

119910119908

0

119861 (Δ119910 | 119910119908) 119889Δ119910 +

⟨119876⟩

119873ch (27)



119861 (119902inv) = 1198861199022

inv119890minus1199022

inv21205902

(28)


++119896





77GeV 196GeV

27GeV 39GeV 624GeV

115GeV

B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06

DataShuffled

200GeV

Δ120578

B(Δ

120578)

0

02

04

06

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18







DataShuffled

DataShuffled

DataShuffled

B(Δ

120601)

Δ120601





0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

0 1 2Δ120601

0 1 2Δ120601

0 1 2 3

3

3

3

3

3

3

3

3Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2

Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2



120594119886119887=⟨119876

119886119876119887⟩

119881 (29)



120594QGP119886119887

= Δ119886119887(119899

119886+ 119899

119886) (30)

where 119899119886 119899



120594HG119886119887= sum

120572

119899120572119902120572119886119902120572119887 (31)

where 119902120572119886



119866120572120573(120578) = 4sum

119886119887119888119889

⟨119899120572⟩ 119902

120572119886120594(had)(minus1)119886119887

(0) 119892(had)119887119888

sdot (120578) 120594(had)(minus1)119888119889

(120578) 119902120573119889⟨119899

120573⟩

(32)


119861120572120573(Δ120578) =

119866120572120573(Δ120578)

119899120573+ 119899

120573

(33)





DataShuffled

DataShuffled

DataShuffled

Kaons

B(q

inv)

1205942ndf = 174838

120590 = 0501

1205942ndf = 807638

120590 = 0504

1205942ndf = 65438

120590 = 0518

1205942ndf = 251638

120590 = 0496

1205942ndf = 753538

120590 = 0509

1205942ndf = 680438

120590 = 0526

1205942ndf = 482238

120590 = 0503

1205942ndf = 963838

120590 = 0519

1205942ndf = 445938

120590 = 0530




03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1 2

2

2

2

2

2

2

2

2

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1



119861 (120601 Δ120601) =1

2


++120601 Δ120601

119873+(120601)

+Δ



119873minus(120601)

(34)





⟨Δ120578⟩ =sum

119894119861 (Δ120578

119894) Δ120578

119894

sum119894119861 (Δ120578

119894) (35)



0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs




+minus(Δ120578 | 120578max)

should be corrected


Δ120578

120578max) (36)





119902 in which


119872(119894 119895) and 119862



= 2119878ℎ+ 1 can be



119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895


119902119894

and 119887119902119895


119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)





119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)


119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)



77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119902119902

119899119902(119901

2) = int119889

3

1199011119899pair119902119902(119901

1 119901

2) (12)





0 Thus the term


0


119899lowast

119902(119901

lowast

) = 119899th (119901lowast

) =119890minus119864lowast119879

412058711989821198791198702(119898119879)

(13)





119872(Δ119910120575119910)

sum

119894=1

119899119886119894(120575119910) equiv 119873

119886(Δ119910) (14)


119899119886119894(120575119910) asymp int

119910119894+1205751199102

119910119894minus1205751199102

1205881119899119886(119910) 119889119910 (15)




defined as [49]

119861119870(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus 1198861198871

sum119872

119894=1119899119886119894

119872(Δ119910120575119910)minus119896

sum

119894=1

119899119886119894sdot (119899

119887(119894+119896)minus 120575

1198861198871205751198960)

119861119870(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus 1198861198871

sum119872

119894=1119899119886119894

119872(Δ119910120575119910)

sum

119894=1minus119896

119899119886119894sdot (119899

119887(119894+119896)minus 120575

1198861198871205751198960)

(16)



119875119894(119886) =

119899119886119894(119899

119887minus 120575

119886119887)

119873119886(119873

119887minus 120575

119886119887) 119875

119894119895(119886119887) =

119899119886119894(119899

119887119895minus 120575

119886119887120575119894119895)

119873119886(119873

119887minus 120575

119886119887)

(17)


1198751198951(119886119887 Δ119910 120575119910) equiv

119875119894119895(119886119887)

119875119894(119886)

=119899119886119894(119899

119887119895minus 120575

119886119887120575119894119895)

119899119886119894(119873

119887minus 120575

119886119887) (18)


119886119894(119899

119887119895minus

120575119886119887120575119894119895)


119875119896Δ119910(119886119887) equiv

119873119896(119886119887 Δ119910 120575119910)

119873119886(Δ119910) (119873

119887(Δ119910) minus 120575

119886119887)

119861119896(Δ119910 120575119910)

equiv1

2

+

sum

119886119887=minus

minus119886119887119875

119896Δ119910(119886119887)119873

119886(Δ119910) (119873

119887(Δ119910) minus 120575

119886119887)

119873119886(Δ119910)

119861119896(Δ119910 120575119910)

equiv1

2sum

119886

119875119896Δ119910(119886119886) minus

sum119886119887119875119896Δ119910(119886119887)119873

119886(Δ119910)119873

119887(Δ119910)

119873119886(Δ119910)

(19)



times10minus3

1

05

0

minus05

minus1

Most central ()70 60 50 40 30 20 10 0



⟨cos

(120601120572+120601120573minus2Ψ

RP)⟩



120574+minus= 119865

119876((sum

119894

cos 2120601119894⟨cos (Δ120601balance)⟩ (120601119894)



)

(20)


119894119901119894

119909= 0 sum

119894119901119894

119910= 0



119894

119905cos(120601


be written as [29]

120574 = minus119865119901

sum119894(cos2120601

119894minus sin2120601

119894)

1198722

tot (21)


119872++ 119872

minus+ 119872





0 05 10

01

02

03

04

05

06

07

minus05 lt y lt 05

0 lt y lt 1

1 lt y lt 2

15 lt y lt 25

B(Δ

y|yw)

Δy

(a)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

B(Δ

y|yw)

Δy

(b)

Bs(Δy)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

Δy

(c)











119861 (Δ119910 | 119910119908) = 119861 (Δ119910 | infin) (1 minus Δ119910) (22)






Pions

0

02

04

06 K0s

1205880

0 04 08 12 16

B(q

inv)

((G

eVc

)minus1)

qinv (GeVc)

(a)

Kaons02

01

0

0 04 08 12 16

120593

qinv (GeVc)

B(q

inv)

((G

eVc

)minus1)

(b)





⟨(1205751198762

)⟩ = ⟨1198762

⟩ minus ⟨119876⟩2

= 1199022

(⟨1198732

⟩ minus ⟨119873⟩2

) (23)

where119876 = 119899+minus119899



⟨119873ch⟩ ⟨1205751198772

⟩ = 4⟨(120575119876

2)⟩

⟨119873ch⟩ (24)

where

119877 =⟨119873

+119873


minus⟩ ⟨119873

+⟩

⟨119873minus⟩ ⟨119873

+⟩

(25)


119863 (119876) = 4⟨(120575119876)

2

⟩

119873ch (26)


119863 (119876)

4= 1 minus int

119910119908

0

119861 (Δ119910 | 119910119908) 119889Δ119910 +

⟨119876⟩

119873ch (27)



119861 (119902inv) = 1198861199022

inv119890minus1199022

inv21205902

(28)


++119896





77GeV 196GeV

27GeV 39GeV 624GeV

115GeV

B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06

DataShuffled

200GeV

Δ120578

B(Δ

120578)

0

02

04

06

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18







DataShuffled

DataShuffled

DataShuffled

B(Δ

120601)

Δ120601





0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

0 1 2Δ120601

0 1 2Δ120601

0 1 2 3

3

3

3

3

3

3

3

3Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2

Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2



120594119886119887=⟨119876

119886119876119887⟩

119881 (29)



120594QGP119886119887

= Δ119886119887(119899

119886+ 119899

119886) (30)

where 119899119886 119899



120594HG119886119887= sum

120572

119899120572119902120572119886119902120572119887 (31)

where 119902120572119886



119866120572120573(120578) = 4sum

119886119887119888119889

⟨119899120572⟩ 119902

120572119886120594(had)(minus1)119886119887

(0) 119892(had)119887119888

sdot (120578) 120594(had)(minus1)119888119889

(120578) 119902120573119889⟨119899

120573⟩

(32)


119861120572120573(Δ120578) =

119866120572120573(Δ120578)

119899120573+ 119899

120573

(33)





DataShuffled

DataShuffled

DataShuffled

Kaons

B(q

inv)

1205942ndf = 174838

120590 = 0501

1205942ndf = 807638

120590 = 0504

1205942ndf = 65438

120590 = 0518

1205942ndf = 251638

120590 = 0496

1205942ndf = 753538

120590 = 0509

1205942ndf = 680438

120590 = 0526

1205942ndf = 482238

120590 = 0503

1205942ndf = 963838

120590 = 0519

1205942ndf = 445938

120590 = 0530




03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1 2

2

2

2

2

2

2

2

2

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1



119861 (120601 Δ120601) =1

2


++120601 Δ120601

119873+(120601)

+Δ



119873minus(120601)

(34)





⟨Δ120578⟩ =sum

119894119861 (Δ120578

119894) Δ120578

119894

sum119894119861 (Δ120578

119894) (35)



0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs




+minus(Δ120578 | 120578max)

should be corrected


Δ120578

120578max) (36)





119902 in which


119872(119894 119895) and 119862



= 2119878ℎ+ 1 can be



119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895


119902119894

and 119887119902119895


119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)





119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)


119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)



77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




times10minus3

1

05

0

minus05

minus1

Most central ()70 60 50 40 30 20 10 0



⟨cos

(120601120572+120601120573minus2Ψ

RP)⟩



120574+minus= 119865

119876((sum

119894

cos 2120601119894⟨cos (Δ120601balance)⟩ (120601119894)



)

(20)


119894119901119894

119909= 0 sum

119894119901119894

119910= 0



119894

119905cos(120601


be written as [29]

120574 = minus119865119901

sum119894(cos2120601

119894minus sin2120601

119894)

1198722

tot (21)


119872++ 119872

minus+ 119872





0 05 10

01

02

03

04

05

06

07

minus05 lt y lt 05

0 lt y lt 1

1 lt y lt 2

15 lt y lt 25

B(Δ

y|yw)

Δy

(a)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

B(Δ

y|yw)

Δy

(b)

Bs(Δy)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

Δy

(c)











119861 (Δ119910 | 119910119908) = 119861 (Δ119910 | infin) (1 minus Δ119910) (22)






Pions

0

02

04

06 K0s

1205880

0 04 08 12 16

B(q

inv)

((G

eVc

)minus1)

qinv (GeVc)

(a)

Kaons02

01

0

0 04 08 12 16

120593

qinv (GeVc)

B(q

inv)

((G

eVc

)minus1)

(b)





⟨(1205751198762

)⟩ = ⟨1198762

⟩ minus ⟨119876⟩2

= 1199022

(⟨1198732

⟩ minus ⟨119873⟩2

) (23)

where119876 = 119899+minus119899



⟨119873ch⟩ ⟨1205751198772

⟩ = 4⟨(120575119876

2)⟩

⟨119873ch⟩ (24)

where

119877 =⟨119873

+119873


minus⟩ ⟨119873

+⟩

⟨119873minus⟩ ⟨119873

+⟩

(25)


119863 (119876) = 4⟨(120575119876)

2

⟩

119873ch (26)


119863 (119876)

4= 1 minus int

119910119908

0

119861 (Δ119910 | 119910119908) 119889Δ119910 +

⟨119876⟩

119873ch (27)



119861 (119902inv) = 1198861199022

inv119890minus1199022

inv21205902

(28)


++119896





77GeV 196GeV

27GeV 39GeV 624GeV

115GeV

B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06

DataShuffled

200GeV

Δ120578

B(Δ

120578)

0

02

04

06

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18







DataShuffled

DataShuffled

DataShuffled

B(Δ

120601)

Δ120601





0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

0 1 2Δ120601

0 1 2Δ120601

0 1 2 3

3

3

3

3

3

3

3

3Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2

Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2



120594119886119887=⟨119876

119886119876119887⟩

119881 (29)



120594QGP119886119887

= Δ119886119887(119899

119886+ 119899

119886) (30)

where 119899119886 119899



120594HG119886119887= sum

120572

119899120572119902120572119886119902120572119887 (31)

where 119902120572119886



119866120572120573(120578) = 4sum

119886119887119888119889

⟨119899120572⟩ 119902

120572119886120594(had)(minus1)119886119887

(0) 119892(had)119887119888

sdot (120578) 120594(had)(minus1)119888119889

(120578) 119902120573119889⟨119899

120573⟩

(32)


119861120572120573(Δ120578) =

119866120572120573(Δ120578)

119899120573+ 119899

120573

(33)





DataShuffled

DataShuffled

DataShuffled

Kaons

B(q

inv)

1205942ndf = 174838

120590 = 0501

1205942ndf = 807638

120590 = 0504

1205942ndf = 65438

120590 = 0518

1205942ndf = 251638

120590 = 0496

1205942ndf = 753538

120590 = 0509

1205942ndf = 680438

120590 = 0526

1205942ndf = 482238

120590 = 0503

1205942ndf = 963838

120590 = 0519

1205942ndf = 445938

120590 = 0530




03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1 2

2

2

2

2

2

2

2

2

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1



119861 (120601 Δ120601) =1

2


++120601 Δ120601

119873+(120601)

+Δ



119873minus(120601)

(34)





⟨Δ120578⟩ =sum

119894119861 (Δ120578

119894) Δ120578

119894

sum119894119861 (Δ120578

119894) (35)



0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs




+minus(Δ120578 | 120578max)

should be corrected


Δ120578

120578max) (36)





119902 in which


119872(119894 119895) and 119862



= 2119878ℎ+ 1 can be



119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895


119902119894

and 119887119902119895


119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)





119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)


119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)



77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 05 10

01

02

03

04

05

06

07

minus05 lt y lt 05

0 lt y lt 1

1 lt y lt 2

15 lt y lt 25

B(Δ

y|yw)

Δy

(a)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

B(Δ

y|yw)

Δy

(b)

Bs(Δy)

21 300

01

02

03

04

05

06

07

08

minus05 lt y lt 05

minus10 lt y lt 10

minus15 lt y lt 15

minus20 lt y lt 20

Δy

(c)











119861 (Δ119910 | 119910119908) = 119861 (Δ119910 | infin) (1 minus Δ119910) (22)






Pions

0

02

04

06 K0s

1205880

0 04 08 12 16

B(q

inv)

((G

eVc

)minus1)

qinv (GeVc)

(a)

Kaons02

01

0

0 04 08 12 16

120593

qinv (GeVc)

B(q

inv)

((G

eVc

)minus1)

(b)





⟨(1205751198762

)⟩ = ⟨1198762

⟩ minus ⟨119876⟩2

= 1199022

(⟨1198732

⟩ minus ⟨119873⟩2

) (23)

where119876 = 119899+minus119899



⟨119873ch⟩ ⟨1205751198772

⟩ = 4⟨(120575119876

2)⟩

⟨119873ch⟩ (24)

where

119877 =⟨119873

+119873


minus⟩ ⟨119873

+⟩

⟨119873minus⟩ ⟨119873

+⟩

(25)


119863 (119876) = 4⟨(120575119876)

2

⟩

119873ch (26)


119863 (119876)

4= 1 minus int

119910119908

0

119861 (Δ119910 | 119910119908) 119889Δ119910 +

⟨119876⟩

119873ch (27)



119861 (119902inv) = 1198861199022

inv119890minus1199022

inv21205902

(28)


++119896





77GeV 196GeV

27GeV 39GeV 624GeV

115GeV

B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06

DataShuffled

200GeV

Δ120578

B(Δ

120578)

0

02

04

06

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18







DataShuffled

DataShuffled

DataShuffled

B(Δ

120601)

Δ120601





0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

0 1 2Δ120601

0 1 2Δ120601

0 1 2 3

3

3

3

3

3

3

3

3Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2

Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2



120594119886119887=⟨119876

119886119876119887⟩

119881 (29)



120594QGP119886119887

= Δ119886119887(119899

119886+ 119899

119886) (30)

where 119899119886 119899



120594HG119886119887= sum

120572

119899120572119902120572119886119902120572119887 (31)

where 119902120572119886



119866120572120573(120578) = 4sum

119886119887119888119889

⟨119899120572⟩ 119902

120572119886120594(had)(minus1)119886119887

(0) 119892(had)119887119888

sdot (120578) 120594(had)(minus1)119888119889

(120578) 119902120573119889⟨119899

120573⟩

(32)


119861120572120573(Δ120578) =

119866120572120573(Δ120578)

119899120573+ 119899

120573

(33)





DataShuffled

DataShuffled

DataShuffled

Kaons

B(q

inv)

1205942ndf = 174838

120590 = 0501

1205942ndf = 807638

120590 = 0504

1205942ndf = 65438

120590 = 0518

1205942ndf = 251638

120590 = 0496

1205942ndf = 753538

120590 = 0509

1205942ndf = 680438

120590 = 0526

1205942ndf = 482238

120590 = 0503

1205942ndf = 963838

120590 = 0519

1205942ndf = 445938

120590 = 0530




03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1 2

2

2

2

2

2

2

2

2

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1



119861 (120601 Δ120601) =1

2


++120601 Δ120601

119873+(120601)

+Δ



119873minus(120601)

(34)





⟨Δ120578⟩ =sum

119894119861 (Δ120578

119894) Δ120578

119894

sum119894119861 (Δ120578

119894) (35)



0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs




+minus(Δ120578 | 120578max)

should be corrected


Δ120578

120578max) (36)





119902 in which


119872(119894 119895) and 119862



= 2119878ℎ+ 1 can be



119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895


119902119894

and 119887119902119895


119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)





119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)


119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)



77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Pions

0

02

04

06 K0s

1205880

0 04 08 12 16

B(q

inv)

((G

eVc

)minus1)

qinv (GeVc)

(a)

Kaons02

01

0

0 04 08 12 16

120593

qinv (GeVc)

B(q

inv)

((G

eVc

)minus1)

(b)





⟨(1205751198762

)⟩ = ⟨1198762

⟩ minus ⟨119876⟩2

= 1199022

(⟨1198732

⟩ minus ⟨119873⟩2

) (23)

where119876 = 119899+minus119899



⟨119873ch⟩ ⟨1205751198772

⟩ = 4⟨(120575119876

2)⟩

⟨119873ch⟩ (24)

where

119877 =⟨119873

+119873


minus⟩ ⟨119873

+⟩

⟨119873minus⟩ ⟨119873

+⟩

(25)


119863 (119876) = 4⟨(120575119876)

2

⟩

119873ch (26)


119863 (119876)

4= 1 minus int

119910119908

0

119861 (Δ119910 | 119910119908) 119889Δ119910 +

⟨119876⟩

119873ch (27)



119861 (119902inv) = 1198861199022

inv119890minus1199022

inv21205902

(28)


++119896





77GeV 196GeV

27GeV 39GeV 624GeV

115GeV

B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06

DataShuffled

200GeV

Δ120578

B(Δ

120578)

0

02

04

06

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18







DataShuffled

DataShuffled

DataShuffled

B(Δ

120601)

Δ120601





0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

0 1 2Δ120601

0 1 2Δ120601

0 1 2 3

3

3

3

3

3

3

3

3Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2

Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2



120594119886119887=⟨119876

119886119876119887⟩

119881 (29)



120594QGP119886119887

= Δ119886119887(119899

119886+ 119899

119886) (30)

where 119899119886 119899



120594HG119886119887= sum

120572

119899120572119902120572119886119902120572119887 (31)

where 119902120572119886



119866120572120573(120578) = 4sum

119886119887119888119889

⟨119899120572⟩ 119902

120572119886120594(had)(minus1)119886119887

(0) 119892(had)119887119888

sdot (120578) 120594(had)(minus1)119888119889

(120578) 119902120573119889⟨119899

120573⟩

(32)


119861120572120573(Δ120578) =

119866120572120573(Δ120578)

119899120573+ 119899

120573

(33)





DataShuffled

DataShuffled

DataShuffled

Kaons

B(q

inv)

1205942ndf = 174838

120590 = 0501

1205942ndf = 807638

120590 = 0504

1205942ndf = 65438

120590 = 0518

1205942ndf = 251638

120590 = 0496

1205942ndf = 753538

120590 = 0509

1205942ndf = 680438

120590 = 0526

1205942ndf = 482238

120590 = 0503

1205942ndf = 963838

120590 = 0519

1205942ndf = 445938

120590 = 0530




03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1 2

2

2

2

2

2

2

2

2

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1



119861 (120601 Δ120601) =1

2


++120601 Δ120601

119873+(120601)

+Δ



119873minus(120601)

(34)





⟨Δ120578⟩ =sum

119894119861 (Δ120578

119894) Δ120578

119894

sum119894119861 (Δ120578

119894) (35)



0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs




+minus(Δ120578 | 120578max)

should be corrected


Δ120578

120578max) (36)





119902 in which


119872(119894 119895) and 119862



= 2119878ℎ+ 1 can be



119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895


119902119894

and 119887119902119895


119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)





119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)


119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)



77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




77GeV 196GeV

27GeV 39GeV 624GeV

115GeV

B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06B(Δ

120578)

0

02

04

06

B(Δ

120578)

0

02

04

06

DataShuffled

200GeV

Δ120578

B(Δ

120578)

0

02

04

06

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18Δ120578

0 06 12 18Δ120578

0 06 12 18

Δ120578

0 06 12 18







DataShuffled

DataShuffled

DataShuffled

B(Δ

120601)

Δ120601





0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

0 1 2Δ120601

0 1 2Δ120601

0 1 2 3

3

3

3

3

3

3

3

3Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2

Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2



120594119886119887=⟨119876

119886119876119887⟩

119881 (29)



120594QGP119886119887

= Δ119886119887(119899

119886+ 119899

119886) (30)

where 119899119886 119899



120594HG119886119887= sum

120572

119899120572119902120572119886119902120572119887 (31)

where 119902120572119886



119866120572120573(120578) = 4sum

119886119887119888119889

⟨119899120572⟩ 119902

120572119886120594(had)(minus1)119886119887

(0) 119892(had)119887119888

sdot (120578) 120594(had)(minus1)119888119889

(120578) 119902120573119889⟨119899

120573⟩

(32)


119861120572120573(Δ120578) =

119866120572120573(Δ120578)

119899120573+ 119899

120573

(33)





DataShuffled

DataShuffled

DataShuffled

Kaons

B(q

inv)

1205942ndf = 174838

120590 = 0501

1205942ndf = 807638

120590 = 0504

1205942ndf = 65438

120590 = 0518

1205942ndf = 251638

120590 = 0496

1205942ndf = 753538

120590 = 0509

1205942ndf = 680438

120590 = 0526

1205942ndf = 482238

120590 = 0503

1205942ndf = 963838

120590 = 0519

1205942ndf = 445938

120590 = 0530




03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1 2

2

2

2

2

2

2

2

2

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1



119861 (120601 Δ120601) =1

2


++120601 Δ120601

119873+(120601)

+Δ



119873minus(120601)

(34)





⟨Δ120578⟩ =sum

119894119861 (Δ120578

119894) Δ120578

119894

sum119894119861 (Δ120578

119894) (35)



0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs




+minus(Δ120578 | 120578max)

should be corrected


Δ120578

120578max) (36)





119902 in which


119872(119894 119895) and 119862



= 2119878ℎ+ 1 can be



119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895


119902119894

and 119887119902119895


119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)





119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)


119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)



77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




DataShuffled

DataShuffled

DataShuffled

B(Δ

120601)

Δ120601





0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

B(Δ

120601)

0

02

04

0 1 2Δ120601

0 1 2Δ120601

0 1 2 3

3

3

3

3

3

3

3

3Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2

Δ120601

0 1 2Δ120601

0 1 2Δ120601

0 1 2



120594119886119887=⟨119876

119886119876119887⟩

119881 (29)



120594QGP119886119887

= Δ119886119887(119899

119886+ 119899

119886) (30)

where 119899119886 119899



120594HG119886119887= sum

120572

119899120572119902120572119886119902120572119887 (31)

where 119902120572119886



119866120572120573(120578) = 4sum

119886119887119888119889

⟨119899120572⟩ 119902

120572119886120594(had)(minus1)119886119887

(0) 119892(had)119887119888

sdot (120578) 120594(had)(minus1)119888119889

(120578) 119902120573119889⟨119899

120573⟩

(32)


119861120572120573(Δ120578) =

119866120572120573(Δ120578)

119899120573+ 119899

120573

(33)





DataShuffled

DataShuffled

DataShuffled

Kaons

B(q

inv)

1205942ndf = 174838

120590 = 0501

1205942ndf = 807638

120590 = 0504

1205942ndf = 65438

120590 = 0518

1205942ndf = 251638

120590 = 0496

1205942ndf = 753538

120590 = 0509

1205942ndf = 680438

120590 = 0526

1205942ndf = 482238

120590 = 0503

1205942ndf = 963838

120590 = 0519

1205942ndf = 445938

120590 = 0530




03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1 2

2

2

2

2

2

2

2

2

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1



119861 (120601 Δ120601) =1

2


++120601 Δ120601

119873+(120601)

+Δ



119873minus(120601)

(34)





⟨Δ120578⟩ =sum

119894119861 (Δ120578

119894) Δ120578

119894

sum119894119861 (Δ120578

119894) (35)



0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs




+minus(Δ120578 | 120578max)

should be corrected


Δ120578

120578max) (36)





119902 in which


119872(119894 119895) and 119862



= 2119878ℎ+ 1 can be



119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895


119902119894

and 119887119902119895


119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)





119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)


119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)



77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




DataShuffled

DataShuffled

DataShuffled

Kaons

B(q

inv)

1205942ndf = 174838

120590 = 0501

1205942ndf = 807638

120590 = 0504

1205942ndf = 65438

120590 = 0518

1205942ndf = 251638

120590 = 0496

1205942ndf = 753538

120590 = 0509

1205942ndf = 680438

120590 = 0526

1205942ndf = 482238

120590 = 0503

1205942ndf = 963838

120590 = 0519

1205942ndf = 445938

120590 = 0530




03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

B(q

inv)

03

02

01

0

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1 2

2

2

2

2

2

2

2

2

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1

qinv (GeVc)0 1



119861 (120601 Δ120601) =1

2


++120601 Δ120601

119873+(120601)

+Δ



119873minus(120601)

(34)





⟨Δ120578⟩ =sum

119894119861 (Δ120578

119894) Δ120578

119894

sum119894119861 (Δ120578

119894) (35)



0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs




+minus(Δ120578 | 120578max)

should be corrected


Δ120578

120578max) (36)





119902 in which


119872(119894 119895) and 119862



= 2119878ℎ+ 1 can be



119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895


119902119894

and 119887119902119895


119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)





119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)


119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)



77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

004

008

012

150 200 250 300 350 400

ssusuu

T (MeV)

120594abs




+minus(Δ120578 | 120578max)

should be corrected


Δ120578

120578max) (36)





119902 in which


119872(119894 119895) and 119862



= 2119878ℎ+ 1 can be



119873(ℎ)

119872= 119863

ℎ

119862119872(119894 119895) 119887

119902119894

119873119902119894

119887119902119895

119873119902119895

(37)

where119873119902119894

and119873119902119895


119902119894

and 119887119902119895


119873(ℎ)

119861= 119863

ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873 (119895)119873

119902(119896)

(38)

119873119861

(ℎ) = 119863ℎ

119862119861(119894 119895 119896) 119887 (119894) 119887 (119895) 119887 (119896)119873

119902(119894)119873

119902(119895)119873

119902(119896)

(39)

where119873119902(119894)

and119873119902(119894)





119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(40)


119901 = 1198861199011199023

Λ | Σ = 119886Λ1199022

119904

Ξ = 119886Ξ119902119904

2

Ω = 119886Ω1199043

(41)



77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




77GeV

39GeV

624GeV200GeV115GeV

B(Δ120578)

B(Δ120601)

Npart

0 100 200 300

Npart

0 100 200 300

⟨Δ120578⟩

⟨Δ120601⟩

052

058

064

09

12

15

STARUrQMD

⟨Δ120578⟩

⟨Δ120601⟩

B(Δ120578) central

B(Δ120601) central

10210

10210

052

058

064

06

09

12

15

radicsNN (GeV)

radicsNN (GeV)



119906119906 + 119889119889 997888rarr 119906119889 + 119889119906 (42)


120588 (119901 119901) = int1198891198751119889119875

2120588119888(119875

1) 120588

119888(119875

2)

sdot int 1198891198751199061

1198891198751199061

1198891199011198892

1198891199011198892

119891 (1198751 119901

1199061

)

sdot 119891 (1198751 119901

1199061

) 119891 (1198752 119901

1198892

)

sdot 119891 (1198752 119901

1198892

) 120575[

[

119901+minus

(1199011199061

+ 1199011198892

)

2

]

]

sdot 120575[

[

119901minusminus

(1199011199061

+ 1199011198892

)

2

]

]

119866119898(119901

1199061

minus 1199011198892

)

sdot 119866119898(119901

1198891

minus 1199011199062

)

(43)

where 1198751and 119875





80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




80

⟨Δ120578⟩

07

06

05



(a)

80


⟨Δ120593⟩

(deg

)

80

60

40


(b)


⟨Δ120578⟩⟨Δ120578⟩

perip

hera

l

⟨Npart⟩

11

1

09

08

07

0 100 200 300 400



(a)


⟨Δ120593⟩⟨Δ

120593⟩ p

erip

hera

l

⟨Npart⟩

0 100 200 300 400

1

08

06

(b)



120588 (119901 119901) = int1198891198751119889119875

2119889119875

3120588119888(119875

1) 120588

119888(119875

2) 120588

119888(119875

3)

sdot int 1198891198751119889119875

2119889119875

3119889119901

1119889119901

2119889119901

3

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 119891 (1198751 119901

1) 119891 (119875

2 119901

2) 119891 (119875

3 119901

3)

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 120575 [119901 minus(119901

1+ 119901

2+ 119901

3)

3]

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

sdot 119866119861(119901

1minus 119901

2 119901

2minus 119901

3 119901

3minus 119901

1)

(44)



119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119902(119894) and 119873

119902(119894)

respectively

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

119873119902(119894) = sum

119894

119873119891

sum

119895=1

119873119891

sum

119896=1

(1 + 120575119894119895+ 120575

119894119896)119863

ℎ

119862119861(119894 119895 119896)

times 119887 (119894) 119887 (119895) 119887 (119896)119873119902(119894)119873

119902(119895)119873

119902(119896)

+sum

ℎ

119873119891

sum

119895=1

119863ℎ

119862119872(119894 119895) 119887 (119894) 119887 (119895)119873

119902(119894)119873

119902(119895)

(45)



119861 (Δ 119884) =1

2

⟨119873+minus(Δ)⟩ minus ⟨119873

++(Δ)⟩

119873+

+119873


minusminus(Δ)⟩

119873minus

(46)



1| = Δ at 119884 sim 119884




119861 (Δ 119884) = 119861119877(Δ 119884) + 119861

119873119877(Δ 119884) (47)



120588119877rarr120587

+120587minus =

119887120587120587

1198732

120575(4)

(119901 minus 1199011minus 119901

2) (48)

where 119901 1199011 and 119901




1198732is given by [38]

1198732= int

11988931199011

1198641

11988931199012

1198642

120575(4)

(119901 minus 1199011minus 119901

2) (49)



119861 (1198752| 119875

1) equiv1

2

119873+minus(119875

1 119875

2) minus 119873

++(119875

1 119875

2)

119873+(119875

1)

+119873

minus+(119875

1 119875

2) minus 119873

minusminus(119875

1 119875

2)

119873minus(119875

1)

(50)

where 1198751and 119875






05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




05 1 15 2 25

120575

B(120575)

c = 0ndash1004

03

02

01

times040

(a)

05 1 15 2 25

120575

c = 10ndash40

B(120575)

04

03

02

01

times044

(b)

05 1 15 2 25

120575

c = 40ndash70

B(120575)

04

03

02

01

times050

(c)

05 1 15 2 25

120575

c = 70ndash96

B(120575)

04

03

02

01

times051

(d)



2gives


1






119909 119877

119910 119879

1205880 120588

2 119886

119904 120591

0 and Δ120591 where 119877

119909 119877




2are the radial


119909and 119877

119910




(51)

where Δ119910 = 1199101minus 119910

2and 119910




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




y)

0 1 2 3 4 5 6

Δy



05

04

03

02

01

0


Eventplane

120601b

120601s




119889119873+minus

119877

1198891199101119889119910

2

= int1198891199101198891199012

perp


1119889119901

perp

2119862120587(119889119873

119877

1198891199101198891199012perp

)120588119877rarr120587

+120587minus (119901 119901

1 119901

2)

(52)


119889119873+minus

119873119877

1198891199101119889119910

2

= 119860int1198891199011

perp119889119901

2

perp119862120587

timesint119889Σ (119909) 1199011sdot 119906 (119909) 119891

120587

119873119877(119901

1sdot 119906 (119909)) 119901

2sdot 119906 (119909)

sdot119891120587

119873119877(119901

2sdot 119906 (119909))

(53)

B(120575)

120575

035

03

025

02

015

01

005

Nonresonance pions


05 1 15 2 25



119861119877(Δ119910)

=1

119873120587

sum

119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

119861119873119877(Δ119910)

=1

119873120587

sum

119873119877

int1198891199101119889119910

2119862120587(119889119873

+minus

119873119877

1198891199101119889119910

2

)120575 (10038161003816100381610038161199102 minus 1199101

1003816100381610038161003816 minus 120575119910)

(54)

in which 119873120587= (119873

120587+ + 119873




1198641198893119873

1198893119901=1

2120587

1198892119873

119901119905119889119901

119905119889119910[1 + 2

infin

sum

119899=1

V119899cos (119899120601 minus ΨPR)] (55)








00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00

002

004

006

008

01

012

014

016

B(Δ

120601)


Δ120601


Data 375∘ lt 120601 lt 525∘

Data 825∘ lt 120601 lt 975∘


Model 375∘ lt 120601 lt 525∘

Model 825∘ lt 120601 lt 975∘



119865 (119883 119875) = 119865 (119903 120601119904 119905 119901

119879 120601

119901 119898)

= Θ(1 minus(119903 cos (120601

119904))

2

(119877119909)2

minus(119903 sin (120601

119904))

2

(119877119910)2

)

sdot 1198701[(119903 120601

119904 119901

119879)] 119890

120573(119903120601119904119901119879) cos(120601

119887120601119901)

119890minus11990521205912

(57)


119904and 120601




1and 119896

2and


04

03

02

01

0000 05 10 15 20

Δy

B(Δ

y)


CentralBlast wave



120588 (120578) =1

2 (cosh 120578)2 (58)





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Au+Au HIJING

p+p HIJING

⟨Δ120578⟩

Npart

0 100 200 30005

06

07

(a)

100 200 300

05

06

0Npart

⟨Δy⟩




(b)











⟨Δ119910⟩ =int119910119908

0119861 (Δ119910119910

119908) Δ119910119889 120575119910

int119910119908

0119861 (Δ119910 | 119910

119908) 119889Δ119910

(59)



Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Pions

Kaons

02

03

04

05

05

01

01

02

03

04

100 200 3000

100 200 3000


120590(G

eVc

)120590

(GeV

c)

radic2m120587Tkin

Npart

Npart



radic2mKTkin

UrQMD Au+Au
















120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














References



























=


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




120578⟩



08

06

04



(a)


⟨Δ120593⟩

(deg

)

0 20 40 60 80



80

60

40

(b)



B(Δ

y)

Δy

0

01

03

02

0 04 08 12 16 2



nch

200GeV130GeV

64GeV22GeV

400 10 20 30

14

12

10

08

06

⟨120575y⟩ Y

119882














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


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



ReviewArticle Balance Function in High-Energy Collisions · are now in order. The rapidity...

Documents

Transcript of ReviewArticle Balance Function in High-Energy Collisions · are now in order. The rapidity...