Conservation Laws and the Multiplicity Evolution of Spectra at the Relativistic Heavy Ion Collider

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by  Zbigniew Chajȩcki

ConservationLaws and theMultiplicity EvolutionofSpectra atthe RelativisticHeavyIonCollider
Zbigniew Chaje¸cki and Mike Lisa†
∗
Department ofPhysics, OhioStateUniversity, 191WestWoodruff Ave, Columbus, OH43210, USA
(Dated:January9,2009)
Transversemomentumdistributionsinultra-relativisticheavyioncollisionscarryconsiderableinformation
about the dynamics of thehot system produced. Direct comparison with the same spectra from p+p colli-
sionshasprovedinvaluabletoidentifynovelfeaturesassociatedwiththelargersystem, inparticular, the“jet
quenching”athighmomentumandapparentlymuchstrongercollectiveflowdominatingthespectralshapeat
lowmomentum. Wepointoutpossiblehazardsofignoringconservationlawsinthecomparisonofhigh-and
9
low-multiplicityfinalstates.Wearguethattheeffectsofenergyandmomentumconservationactuallydominate
0
manyoftheobservedsystematics,andthat p+pcollisionsmaybemuchmoresimilartoheavyioncollisions
0
thangenerallythought.
2

n
a I. INTRODUCTION B. Biggerisbetter
J

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Despite the necessary attention to smaller colliding part-
A. HeavyIonPhysics:RelyingonComparison
] ners, these comparisons are ultimately aimed at identifying
h novelaspectsofcollisionsbetweentheheaviestions,inwhich
t
- The physics program at the Relativistic Heavy Ion Col- ahighlyexcitedbulksystemmightbecreated,withasufficient
cl lider (RHIC) at Brookhaven National Laboratory is remark- numberof degreesof freedomsuch that it may be described
u ably rich, thanks to the machine’s unique ability to collide thermodynamically– e.g. in terms of pressure, temperature,
n nucleifrom1Hto197Au,infullysymmetric(e.g. Au+Auor energydensity,andanEquationofState(EoS).Iftheenergy
[ p+p ) to strongly asymmetric (e.g. d+Au) entrance chan- densityofthissystemissufficientlylarge(typicallyestimated
2 nels, over an energy range spanning more than an order of at e crit 1 GeV/fm3 [6]) and its spatial extent considerably
∼
v magnitude. The capability to collide polarized protons pro- largerthan the color-confinementlength 1 fm, then a new
∼
9 vides access to an entirely new set of fundamental physics, stateofmatter–thequark-gluonplasma(QGP)[10]–maybe
6 notdiscussedfurtherhere. created. Microscopically, such a state might be character-
5
izedbycoloredobjects(orsomethingmorecomplicated[11]);
3 Achieving the primary aim of RHIC– the creation and
macroscopically,it representsa regionon the phase diagram
. characterizationof a color-deconfinedstate of matter and its
7 in whichthe EoSis distinctly differentthanfor the hadronic
0 transition back to the confined (hadronic)state– requiresthe phase[12].
8 full capabilities of RHIC. In particular, comparisons of par-
Ultra-relativisticcollisionsbetweentheheaviestnucleien-
0 ticle distributions at high transverse momentum (p ) from
T joytheadditionaladvantagethatfinite-sizeeffectsaresmall,
v: Au+Au and p+p collisions, probe the color-opaquenature due to high-multiplicity final states. In a small system (e.g.
i ofthehotsystemformedinthecollisions[1,2,3]. Compari-
X final state of an p+ p¯ collision) a statistical analysis of
sonwithreferenced+Acollisionswerenecessarytoidentify
yields requires a canonical treatment, due to the conserva-
r the role of initial-state effects in the spectra [4]. Comparing
a tionofdiscreetquantumnumberssuchasbaryonnumberand
anisotropic collective motion from non-central collisions of
strangeness[13]. Forlargersystems,agrandcanonicaltreat-
different-massinitialstates(e.g. Au+AuversusCu+Cu)[5]
mentismorecommon[e.g.14],withfinitequantum-number
teststhevalidityoftransportcalculationscrucialtoclaimsof
effectsabsorbedinto,e.g.“saturationfactors”[15].
thecreationofa“perfectliquid”atRHIC[6]. Indeed,amain
Duetothelargeavailableenergy√sandfinal-statemulti-
componentofthefutureheavyionprogramatRHICinvolves
plicity, energy and momentum conservation effects on kine-
adetailedenergyscan,designedtoidentifyapredictedcritical
matic observables (spectra, momentum correlations, elliptic
pointintheEquationofStateofQCD[7].
flow)aregenerallysmall. Theyareaccountedforwithcorrec-
The need for such systematic comparisons is not unique tionfactors[16,17]orneglectedaltogether.
to RHIC, but has been a generic feature of all heavy ion
programs [8, 9], from low-energy facilities like the NSCL
(Michigan State), to progressively higher-energy facilities
C. Multiplicityevolutionofsingle-particlespectra
at SIS (GSI), the Bevatron/Bevalac (Berkeley Lab), AGS
(Brookhaven), and SPS (CERN). The nature of heavy ion
physicsissuchthatlittleislearnedthroughstudyofasingle Detailedsingle-particlespectra(e.g. d2N/dp2T)havebeen
system. measuredatRHIC,foravarietyofparticletypes. Often,the
shape of the “soft” (p .2 GeV/c) part of the spectrum is
T
comparedtohydrodynamiccalculations[18]orfittedtosim-
ple“blast-wave”parameterizations[e.g.19]toextractthecol-
lectiveflowofthesystem. The“hard”sector(p &4GeV/c)
∗Electronicaddress:[email protected] T
†Electronicaddress:[email protected] isassumedtobedominatedbythephysicsoftheinitial-state,
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high-Q2 parton collisions and resulting jets. The physics of of Fermi’s Golden Rule, in which dynamics and kinematics
the“firm”sector(2.p .4GeV/c)maybetherichestofall, (phasespace) factorize. This leads to a formula for finite-
T
reflectingthedynamicsoftheconfinementprocessitself[20]. numbereffectsonsingle-particlespectra, duesolely to kine-
We wouldlike tofocusnotso muchon thesingle-particle matics,forafixeddynamical(“parent”)distribution.
spectra themselves, but on their multiplicity dependence. In Section III, we test the extreme ansatz that all of the
Much has been inferred from this dependence. In the soft experimentally-measured multiplicity dependence of single-
sector, blast-wave fits to spectra from high-multiplicity final particle spectra is due to EMCICs. We will find surprising
states(associatedwithcentralA+Acollisions)indicatestrong agreementwiththisansatzinthesoftsector(pT .1GeV/c).
collective radial flow; the same fits to low-multiplicity final We will discuss that our formalism is on less firm footing,
states– including minimum bias p+p collisions– appear to conceptually and mathematically, at much higher pT. Nev-
indicate much weaker flow [21]. This seems to confirm a ertheless, we explore this regime as well. We find that, in
commonassumptionthat p+pcollisionsarenotsufficiently the hard sector, the data from heavy ion collisions is clearly
“large”todevelopbulkcollectivebehaviour. notdominatedbyEMCICs,thoughwepointoutthatignoring
Inthehardsector,oneoftheearliestandmostexcitingob- EMCICs, especially for p+p collisions, may be dangerous
servations [3, 22] at RHIC was that the high-pT yield from evenathigh pT.
high-multiplicityAu+Aucollisions was suppressed, relative In Sections IV and V, we summarize and give an outlook
toappropriatelyscaledlower-multiplicityA+Aorminimum forfuturestudies.
bias p+pcollisions. Thishasbeentakenasevidenceofen-
ergylossofhard-scatteredpartonsthroughaverycolor-dense
medium. Meanwhile, the high-p partofthe spectrumfrom II. EFFECTSOFENERGYANDMOMENTUM
T
high-multiplicityp+pcollisionsappearenhancedrelativeto CONSERVATIONONSINGLE-PARTICLESPECTRA
low-multiplicity p+pcollisions[23],againsuggestingthata
color-densebulksystemisnotproducedin p+pcollisions. A. Arestrictedphasespacefactor
In this paper, we discuss the effects of energy and
momentum conservation on the multiplicity evolution of Changingthesize(centralversusperipheralioncollisions,
single-particle spectra at RHIC. Energy and momentum e+ecollisions,etc)andenergyofacollisionsystemwilllead
conservation-induced constraints (EMCICs) [64] have been to different measured single-particle distributions, reflecting
largelyignoredin the analysesjust mentioned,probablydue (1) possibly different physical processes driving the system
to two reasons. The first is the field’s usual focus on the and (2) effects due to phase space restrictions. To focus
highest-multiplicity collisions, where such effects are as- on changes caused by the latter, we consider some Lorentz-
sumed small; it seems natural to compare analyses of such invariant“parent”distribution f˜(p) 2Ed3N,drivenbysome
systemsto“identical”onesofsmallersystems,forgettingthat ≡ dp3
unspecified physical process, but unaffected by energy and
EMCICeffectsplayanever-increasingroleinthelattercase.
momentumconservation. For simplicity, we assume that all
Perhaps the more important reason is that EMCICs do not
particlesobeythesameparentdistribution.
generate “red flag” structures on single-particle spectra; this
In the absence of other correlations, the measured single-
is in contrastto multi-particlecorrelationanalyses, in which
particle distributionis related to the parentaccordingto [16,
conservationlaw-inducedcorrelationsmaybemanifestlyob-
17,25,27]
viousandhaveevenbeenusedtoestimatethenumberofun-
measuredneutralparticlesinhighenergycollisions[24]. Es- f˜ (p )= f˜(p ) (1)
c 1 1
pecially with the enhanced attention on precision and detail ×
at the SPS and RHIC, there has been increasing discussion (cid:213) N d4p d p2 m2 f˜(p ) d 4 (cid:229) N p P
j=2 j j− j j i=1 i−
of EMCIC effects in 2-particle [17, 25], 3-particle [26], and R ,
N-particle[27]observables. Below,weshowthatEMCICef- (cid:16)(cid:213) Nj=1d4pjd (cid:16)p2j−m2j(cid:17)f˜(pj)(cid:17)d 4(cid:0)(cid:229) Ni=1pi−P(cid:1)
fects on single-particle spectra are also significant, and may R
(cid:16) (cid:16) (cid:17) (cid:17) (cid:0) (cid:1)
evendominatetheirmultiplicityevolution. whereN istheeventmultiplicity. Theintegralinthenumer-
atorofEquation1representsthenumberofconfigurationsin
which the N 1 other particles counter-balance p so as to
1
−
D. Organizationofthispaper conservethe totalenergy-momentumP of the event,and the
denominator,integratingoverall N particles, isa normaliza-
tion.
Several authors [e.g. 28] have discussed finite-number ef-
ForN &10[25],onemayusethecentrallimittheoremto
fects in statistical models, and many numerical simulations
rewritethefactorinEquation1as[16,17,25,27]
ofsubatomiccollisionsconserveenergyandmomentumauto-
matically[e.g.29,30].However,aspointedoutbyKnoll[31],
N 2
our question– to what extent do EMCICs alone explain the f˜ (p)= f˜(p) (2)
c i i
multiplicityevolutionofspectra?–cannotbeaddressedfrom · N 1 ×
(cid:18) − (cid:19)
these simulations themselves, since dynamic and kinematic 1 p2 p2 p2 (E E )2
i,x i,y i,z i
evolutionareinterwoveninthesemodels. Thus,inSectionII, exp + + + −h i ,
we discuss a formalism based on Hagedorn’s generalization "−2(N−1) hp2xi hp2yi hp2zi hE2i−hEi2!#
3
where volume required to describe the data was considered unre-
alistically large [36]. Using the mean value theorem, Hage-
pn dpf˜(p) pn (3) dorn[33]generalizedthetheorysothatthe“physicsterm”is
h µi≡Z · µ
theinteractionmatrixelement,suitablyaveragedoverallfinal
states.
are average quantities and we have set the average three-
We wish to make no assumptions about the underlying
momentum p =P /N=0. Westressthatwhat
appearsinEhqu(aµt=io1,n2,33)iistheµp=a1r,e2,n3tdistribution f˜,notthemea- physics(representedby f˜)drivingtheobservedspectrum f˜c.
sured one f˜. Hence, for finite multiplicity N, the averages Rather,wewishtoquantifytheeffectofchangingthemulti-
c plicityN,whichappearsinthephasespaceterm.
pn arenotthemeasuredones,whichwedefineas
h µi In particular, in the following Section, we compare mea-
suredsingle-particlespectrafordifferenteventclasses.
hpnµic≡Z dpf˜c(p)·pnµ. (4) Wepostulatethattheparentdistributionsfor,sayclasses1
and 2, are the same (f˜ = f˜). By Equation 3, this implies
1 2
SeealsothediscussioninAppendixB. p = p p . In this case, the only reason that the
µ 1 µ 2 µ
Since pT distributions are commonly reported, we would ohbsiervedhspeictr≡adhiffeir(f˜ = f˜ )isthedifferencein“mul-
c,1 c,2
like to estimate EMCIC distortionsto p distributions, inte- 6
T tiplicity”N =N ;seeSectionIICforadiscussionofN .
1 2 1
gratedoverazimuthandafiniterapiditybincenteredatmidra- 6
To eliminate the (unknown) parent distribution itself, we
pidity. As discussed in Appendix A, for the approximately
will study the ratio of observed p distributions, which, by
T
boost-invariantdistributionsatRHIC[21], the measuredand
Equation5becomes
parentp distributionsarerelatedby
T
f˜ (p ) (N 1)N 2
N 2 c,1 T =K 2− 1 (6)
f˜c(pT)= f˜(pT)· N 1 × (5) f˜c,2(pT) ×(cid:18)(N1−1)N2(cid:19) ×
(cid:18) − (cid:19) 1 1 2p2
1 2p2 p2 exp T +
exp"−2(N−1) hp2TTi+hp2zzi (cid:20)(cid:18)2E(2N2−1)−22(EN1E−1)(cid:19)(cid:18)hp2TEi 2
+ E2 E2 E 2− E22EhEEi 2+ E2hEi2E 2!#. +hE2i−hEi2 −hE2i−hhiEi2+hE2hi−ihEi2!#,
h i−h i h i−h i h i−h i
wherethe constantK is discussed atthe endof Section IIC.
ThenotationX indicatestheaverageofaX overtherapidity As mentioned at the end of Section IIA, numerically unim-
intervalused;seeAppendixAfordetails. Theseaveragesde- portanttermsin pz havebeendropped.
pend,ofcourse,onp andshouldnotbeconfusedwithglobal Naturally, our postulate cannot be expected to be entirely
T
averages X (Equation3)whichcharacterizethe parentdis- correct; one may reasonablyexpectthe mix of physicalpro-
h i
tribution. cessesin p+pcollisionstodifferfromthoseinAu+Aucol-
We wouldalso like to emphasizethefactthatsince Equa- lisions. Nevertheless, it is interesting to find the degree to
tion5dependsontheenergyoftheparticle(notjustmomen- whichthechangeinsingle-particlespectramaybeattributed
tum) it becomes clear that the EMCIC effects are larger on only to finite-multiplicity effects. We will find that the pos-
heavierparticlesatthesame p . Thusweshouldexpectthat tulate works surprisingly well in some regions, and fails in
T
theprotonspectrawillbemoresuppressedthanpionspectra. others. Aswewilldiscuss,boththesuccessandfailureraise
In what follows, we find that ignoring the p2/ p2 term interestingandsurprisingpossibilities.
z h zi
does not affect our results, since the numerator is small for
thenarrowrapiditywindowsusedhere,andthedenominator
islarge.Indiscussionsbelow,wesetthistermtozero. C. Testingthepostulate-howtotreattheparameters
Byourpostulate,thephasespacefactoraffectinga p dis-
T
B. Straw-manpostulateofauniversalparentdistribution tribution is drivenby four quantities. Three, p2 , E2 and
h Ti h i
E ,characterizetheparentdistribution,whileN isthenum-
h i
Equations 1-5 are reminiscent of Fermi’s “Golden berofparticlesinthefinalstate.Ingeneral,increasinganyone
Rule”[32,33],inwhichtheprobabilityformakingaparticu- parameterdecreasesthe effectof phase space restrictionson
lar observationisgivenbytheproductofthesquaredmatrix the observeddistributions. But whatshouldwe expectthese
elementand a quantitydeterminedbyavailable phasespace. valuestobe? Theyshouldcharacterizetherelevantsystemin
Thefirsttermrepresentedtheunderlyingphysicalprocess. In whichalimitedquantityofenergyandmomentumisshared.
his original statistical model [32], Fermi originally assumed Theyare not, however,directlymeasurable, andshouldonly
it to be a constantrepresentingthe volume in which emitted approximately scale with measured values, for at least five
particleswereproduced;thisisequivalenttosetting f˜(p)con- reasonsdiscussedhere.
stantinEquation1. Whilesurprisinglysuccessfulinpredict- Firstly, the energy and momentum is shared among mea-
ingcrosssectionsandpionspectra[e.g.34,35],theemission sured and unmeasured (neutrals, neutrinos, etc.) particles
4
alikesothatN shouldroughlytrackthemeasuredeventmul- h max hNi hp2Tic hp2zic hE2ic hEic
tiplicity N , but need not be identical to it. Secondly,
meas 1.0 7.5 0.58 0.41 1.45 0.98
emissionofresonancessmearstheconnectionbetweenNand
2.0 13.4 0.59 2.81 3.89 1.57
N ; e.g. the emission of an omegameson which later de-
meas
cays into “secondary” particles (w ppp ) increments N by 3.0 17.9 0.59 12.95 14.01 2.65
unity,ratherthanthree,asfarasothe→rparticlesareconcerned. 4.0 21.5 0.59 82.45 83.55 5.13
This latter consideration also affects the kinematic parame- 5.0 23.4 0.59 262.88 265.03 8.29
ters p2 , E2 and E . While energy and momentum are, ¥ 23.6 0.59 275.23 276.4 8.48
h Ti h i h i
ofcourse,conservedinresonancedecay,theaforementioned
quantities, themselves, are not. Thus, one need not expect TABLE I: For a given selection on pseudorapidity h < h max,
| |
perfectcorrespondencebetweentheappropriatekinematicpa- the number and kinematic variables for primary particles from a
rametersinEquation6,andthemeasuredones. PYTHIAsimulationofp+pcollisionsat√sNN=200GeVaregiven.
UnitsareGeV/cor(GeV/c)2,asappropriate. 100keventswereused
Thirdly,evenrestrictingconsiderationtoprimaryparticles,
andalldecayswereswitchedoffinsimulations.
it is unclearthat all of them should be consideredin the rel-
evant ensemble of particles sharing some energy and mo-
mhigenht-uemne.rgIyncpoallritsiicounlas,r,thfoermsopmaceen-ttiummeeexxtteenntdoefdcshyasrtaecmtesriisn- h max hNi hp2Tic hp2zic hE2ic hEic
1.0 16 0.20 0.11 0.40 0.44
tic physics processes (e.g. string breaking) and causality in
2.0 29 0.21 0.76 1.05 0.68
anapproximatelyboost-invariantscenariosuggestthatrapid-
ity slices of roughly unit extent should be considered sepa- 3.0 39 0.21 3.5 3.8 1.2
rate subsystems [26]. Of course, the total available energy 4.0 47 0.21 24 25 2.2
in any event is shared among all such subsystems; i.e. the 5.0 51 0.22 88 89 3.7
midrapiditysubsysteminoneeventwillnothaveexactlythe
sameavailableenergyasthatinanotherevent.However,such TABLEII:Foragivenselectiononpseudorapidity h <h max, the
| |
fluctuationsaretobeexpectedinanycase–surelyindividual numberandkinematicvariablesforfinalstateparticles(particlein-
collisionswilldifferfromoneanothertosomeextent. Thus, dexKS=1inPYTHIA)fromaPYTHIAsimulationofp+pcollisions
we repeat our interpretation of the four parameters N, p2 , at√sNN =200GeVaregiven. 100keventsweregeneratedandde-
E2 and E : theycharacterizethescale, inenergyandhmToi- faultPYTHIAparameterswereusedinsimulations. UnitsareGeV/c
h i h i or(GeV/c)2,asappropriate.
mentum,ofthelimitedavailablephasespacetoanN-particle
subsystem.
Fourthly,Equations1-6 are appropriatefor fixed N, while
expectations may be set. Table I summarizes the result for
we will be comparing to measured spectra selected by mea-
primary particles satisfying a varying cut on pseudorapidity
suredcharged-particlemultiplicity. Thus,N wouldinevitably
fluctuate within an event class, even if we could ignore the whereallparticledecayswhereswitchedoffin PYTHIA sim-
ulations. The results from simulations when resonance de-
aboveconsiderations.Naturally,highmultiplicityeventscon-
cays were included in simulations are presented in Table II.
tribute to spectra more than low multiplicity events. Simi-
Thesetwotablesgivesusroughestimatesofrangesoftheto-
larly, the average multiplicity in two-particle correlations is
talmultiplicityandkinematicvariablesthatone mayexpect.
evenmoreshiftedtohighermultiplicities.
The bulk component of single-particle spectra is often esti-
Fifthly,asalreadymentionedinSectionIIA,thekinematic
mated with Maxwell-Boltzmann distributions, with inverse
parameters p2 , E2 and E correspondto the parentdis-
h Ti h i h i slope parametersin the range T 0.15 0.35GeV. Again,
tribution, whichwill onlycorrespondidenticallyto the mea- ∼ ÷
simply for rough guidance, we list Maxwell-Boltzmann ex-
suredoneinthelimitofinfinitemultiplicity(i.e. noEMCIC
pectationsforourkinematicparametersinTableIII,assuming
distortions).SeealsothediscussioninAppendixB.
pion-dominatedsystem.
For all of these reasons, we will treat N, p2 , E2 and
h Ti h i Finally,awordaboutnormalization–thequantityK which
E asfreeparameterswhentestingourpostulateagainstdata.
h i appearsinEquation6. Notonlyenergyandmomentum,but
Ouraimisnotto actuallymeasurethesequantitiesbyfitting
also discrete quantum numbers like strangeness and baryon
thedatawithEquation6;thisisgood,sinceourfitstothedata
onlyveryroughlyconstrainourfourparameters,asdiscussed
in the nextSection. Rather, our much less ambitiousgoalis
non-rel.limit ultra-rel. ifT =0.15 0.35GeV
to see whether “reasonable” values of these parameters can ÷
limit
explainthemultiplicityevolutionofthespectra.
p2 2mT 8T2 0.045 0.98(GeV/c)2
To get a feeling for these values, we look at p+p colli- h Ti ÷
sionsat√sNN=200GeV,simulatedbythePYTHIAeventgen- hE2i 145T2+m2 12T2 0.10÷1.50GeV2
erator (v6.319)[37]. In the model, we can identify primary E 3T+m 3T 0.36 1.00GeV
h i 2 ÷
particles, thus avoiding some of the issues discussed above.
However, the fact that PYTHIA conservesmomentummeans TABLE III: The average kinematic variables obtained from the
thatweaccesshpnµicasdefinedbyEquation4,nottheparam- Mrelaaxtiwviesltli-cBaonltdzmulatrnan-redliasttirvibisuttiicolnimfi(t.pA)=pioddnpN3ga∼sies−aEs/sTumuesdin.g non-
eters of the parentdistribution. Nevertheless, a scale for our
5
number are conserved event by event, affecting the overall Eventselection N p2 [(GeV/c)2] E2 [GeV2] E [GeV]
h Ti h i h i
yield of a given particle species. For example, the related
p+pmin-bias 10.3 0.12 0.43 0.61
phenomenon of “canonical suppression” affects the ratio of
Au+Au70-80% 15.2 ” ” ”
yieldsforstrangeversusnon-strangeparticles,asmultiplicity
varies[38, 39]. Sincewe restrictourattentiontoenergyand Au+Au60-70% 18.3 ” ” ”
momentumconservationand the effect on kinematic quanti- Au+Au50-60% 27.3 ” ” ”
ties, we are interested in the shape of the spectra ratio, as a Au+Au40-50% 38.7 ” ” ”
functionofparticlemomentum,andincludeafactorK inour Au+Au30-40% 67.6 ” ” ”
Equation6,whichshouldbeoforder,butnotnecessarilyiden-
Au+Au20-30% 219 ” ” ”
ticalto,unity.Wedonotdiscussitfurther.
Au+Au10-20% >300 ” ” ”
Au+Au5-10% >300 ” ” ”
Au+Au0-5% >300 ” ” ”
III. TESTOFTHEPOSTULATE-COMPARISONTODATA
TABLEIV: Multiplicity and parent-distribution kinematic parame-
Wenowexplorethedegreetowhichthepostulateproposed ters which give a reasonable description of the spectrum ratios for
above describes the multiplicity evolution of measured p identifiedparticlesinthesoftsector. Seetextfordetails. Notethat
T
spectra measured in √sNN = 200 GeV collisions at RHIC. the multiplicitychanges withevent class; theparent distributionis
As is frequently done, we will separately discuss the “soft” assumedidentical.
(p .1 GeV/c) and “hard” (p &3 GeV/c) portions of the
T T
spectra. Thisseparationisnotentirelyarbitrary,asspectrain
A. Softsector:identifiedparticlesinAu+Auversusp+p
thesetwo p rangesarethoughttobedominatedbyquitedif-
T
ferentphysics, and the multiplicity evolutionin the two sec-
tors is usually interpreted in terms of distinct physics mes- Figure 1 shows mT distributions for minimum-bias
sages. p+pcollisionsand multiplicity-selectedAu+Aucollisions,
In the soft sector, the spectral shapes are often consis- allat√sNN =200GeV,reportedbytheSTARCollaboration
tent with hydrodynamic calculations [e.g. 18, 40], or fitted atRHIC[21]. Forthehighest-multiplicityAu+Aucollisions
with blast-wave type models [e.g. 19, 41], and show evi- (top-most filled datapoints), the spectrum for heavier emit-
dence of strong, explosive flow associated with a collective tedparticlesislesssteepthantheessentiallyexponentialpion
bulkmedium.Thisisespeciallyclearinthemassdependence spectrum. Circles in Figure 2 show the result of fits with a
of the spectra; the m (or p ) spectrum of heavy particles blast-wavemodel[19]. Theyindicateakineticfreezeouttem-
T T
likeprotonsaresignificantlyflatterthanthatforpions,inthe peratureofabout100MeVandaveragecollectiveflowveloc-
presence of strong flow. The multiplicity evolution in this ityabout0.6cforthemostcentralcollisions.Forlowermulti-
sector suggests that high-multiplicity collisions (say, central plicitycollisions,thefreeze-outtemperatureappearstogrow
Au+Aucollisions) show much morecollective flow than do to 130 MeV and the flow velocity decreases to 0.25c.
∼ ∼
low-multiplicity (say, p+p ) collisions [21]. Such an inter- TheSTARcollaboration,usingaslightlydifferentimplemen-
pretationinitiallysensibleinascenarioinwhichflowisbuilt tation of a blast-wave model, reported essentially identical
up through multiple collisions among emitted particles; the values[21].
conceptofacollectivebulkmediuminaverylow-multiplicity Ratios of spectra from minimum-bias p+p collisions to
collisionisthususuallyconsideredquestionable. thosefromAu+AucollisionsareplottedinFigure3. Forthe
Particleyieldsathigh p ,ontheotherhand,aregenerally filledpoints,thedenominatoristhemostcentralAu+Aucol-
T
discussed in the context of fragments from high-Q2 parton lisions,whiletheopenpointsrepresenttheratiowhenthede-
scatterings in the initial stage of the collision. As the event nominatorisfromperipheral(60-70%centrality)Au+Aucol-
multiplicity in Au+Aucollisions is increased, a suppression lisions. Pions,kaons,andprotonsaredistinguishedbydiffer-
of high-p yields is observed, relative to a properlynormal- entsymbolshapes.
T
izedminimum-biasspectrumfromp+pcollisions.Thissup- The curvesshowthe functiongivenin Equation6, for the
pressionhasbeenattributedtopartonicenergylossinthebulk kinematic scales given in Table IV. Clear from the Table is
medium[42,43,44,45]. thatall curvesin Figure3 are generatedwith the same kine-
Themultiplicityevolutionofthespectrainp+pcollisions, matic variables p2 , E2 and E ; only the relevantmulti-
h Ti h i h i
however,showsquitethereverse. Relativeto thesoftsector, plicitychanges.
thehigh-pT yieldsincreaseasthemultiplicityincreases;one We do not quote uncertainties on the kinematic or multi-
mayalso saythatthe pT spectrabecomeless steepasmulti- plicityparameters,asthefittingspaceiscomplex,withlarge
plicityincreases[23]. Thisseemstoreinforcetheconclusion correlations between them. Furthermore, it is clear that the
discussed abovein relation to the soft sector, that p+pcol- calculatedcurvesdonotperfectlyreproducethemeasuredra-
lisions do not build up a bulk system capable of quenching tios. However,itisalsoclearthat“reasonable”valuesofmul-
jets. tiplicityandenergy-momentumscalesgoalongwaytowards
Here, we reconsider these conclusions based on the mul- explainingthemultiplicityevolutionofthespectra,evenkeep-
tiplicity evolution of the spectra, in light of the phase space ingphysics(“parentdistribution”)fixed.OurpostulateofSec-
restrictionsdiscussedabove. tionIIBseemstocontainagooddealoftruth.
6
2] p - K- 10 p
V
e
G
4/102 10
c
[ 1
)
y
d
T
m
d10 1
mT 10-1

p2
(
N/ 1
2d 10-1 10-2
10-1
10-2 10-3
0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5 0.6
mT - mp [GeV/c2] mT - mK [GeV/c2] mT - mp [GeV/c2]
FIG.1: (Coloronline)Transversemassdistributionsforpions(left),kaons(center)andantiprotons(right)measuredbytheSTARCollab-
oration for √sNN=200 GeV collisions [21]. The lowest datapoints represent minimum-bias p+p collisions, while the others come from
Au+Aucollisionsofincreasingmultiplicity.Filleddatapointsareforthetop5%and60-70%highest-multiplicityAu+Aucollisions,andfor
thep+pcollisions.
140
BW fit to STAR data u)
A
130 PRL 92 (2004) 112301 +
] u
V A
Me 120 BW fit to STAR data , T
p
[ (EMC corrected) (
c
n f
Tki110 p)/10-2
+
p
100 ,T
p
(
c
0.6 f
0.5
> STAR Au+Au @ 200 GeV
0.4
b< [0 -5]% [6 0-70] % p -
-
0.3 K
p
0.2
0 100 200 300 400 500 600 700 0 0.2 0.4 0.6 0.8 1 1.2
dN /dh p [GeV/c]
ch T
FIG.2: (Coloronline)Circlesshowthetemperature(toppanel)and FIG. 3: (Color online) The ratio of the pT distribution from
flow(bottompanel)parametersofaBlast-wavemodel[19]fittothe minimum-bias p+pcollisionstothedistributionfrom0-5%(filled
STAR spectra of Figure 1, as a function of the event multiplicity. datapoints) and 60-70% (open datapoints) highest multiplicity
Squares represent Blast-wave fit parameters to “EMCIC corrected Au+Aucollisions;c.f. Figure1. Theratioofthekaonspectrafrom
spectra,” andshaded regionrepresents theseresultscombined with p+p and 0-5% Au+Au collisions (solid green squares) has been
systematicerrors,asdiscussedinthetext. scaledbyafactor 1.7forclarity. Curvesrepresent acalculationof
thisratio(ratioofEMCICfactors)usingEquation6.
Another way to view the same results is useful. While
the curves shown in Figure 3 only approximately describe shown in Figure 4, where the measured min-bias p+p and
the data shown there, one may approximately “correct” the centralandmid-peripheralAu+Auspectrahavebeencopied
measured m distributions, to account for EMCICs. This is fromthefullpointsofFigure1andareshownbyfullpoints.
T
7
2] p - K- 10 p
V
e
G 10
4/102
c 1
[
)
y
d 1
T
m
d10 10-1
T
m
10-1
p2
(
N/ 1 10-2
2d Au+Au [60-70]% p+p
10-2 p+p "EMC corrected"
Au+Au [60-70]%
p+p "EMC corrected"
10-1 Au+Au [0-5]% "EMC corrected" 10-3 (scaled to Au+Au)
10-3
0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5
mT - mp [GeV/c2] mT - mK [GeV/c2] mT - mp [GeV/c2]
FIG. 4: (Color online) Transverse mass distributions of pions, kaons and antiprotons for minimum-bias p+p collisions and 60-70% and
0-5% highest multiplicityAu+Aucollisions at √sNN =200 GeV. Filleddatapoints are the same asin Figure 1. Open triangles represent
the p+pspectradividedbythelowercurvesshowninFigure3. Opencirclesarethesamespectraastheopentriangles, exceptscaledup
tocomparetothespectrafromtheAu+Aucollisions. Opensquaresrepresentthespectrafrom60-70%highestmultiplicityAu+Auevents,
dividedbytheratioofupperandlowercurvesshowninFigure3.Seetextfordetails.
The open red triangles represent the min-bias p+p spectra, extracted via our approximate EMCIC correction procedure
divided by Equation 6, with the parameters from Table IV. followtheBlast-waveshapeonlyapproximately.Muchofthe
This“EMCIC-corrected”spectrumisthenscaleduptoshow deviation is at p 0.9 GeV/c for protons from the lowest
T
∼
comparisontothespectrafromcentralAu+Au(openredcir- multiplicitycollisions(upper-rightpanels). Thisistheregion
cles); the levelof(dis)agreementisidenticalto thatbetween around which the approximations used in deriving the EM-
thelowerdatapointsandcurvesinFigure3. CIC correction should start to break down, as discussed in
Spectra from the mid-central Au+Au collisions have Appendix B. So, two fits are performed: one including all
been likewise “corrected.” The open squares in Figure 4 datapoints shown (blue squares in Figure 2), and the other
may be compared to the open circles; again the level of excluding proton spectra points with pT >0.8 GeV/c. The
(dis)agreementis equivalentto that between the upper data- resulting range of Blast-wave parametersis indicated by the
pointsandcurvesinFigure3. shaded regionin Figure 2. There, statistical errorson the fit
Spectra themselves contain more information than two- parametershavebeenmultipliedby c 2/d.o.f.(rangingfrom
parameter fits to spectra. However, much has been made of 2 for spectra from p+p collisions to 1 for those from
∼ p ∼
blast-wavefitstomeasuredp spectra,whichsuggestamuch mid-peripheral and central Au+Au collisions) and added to
T
largerflowincentralAu+Aucollisions,relativeto p+pcol- both endsof the range. Thus, the shaded region shouldrep-
lisions. Thus, it may be instructive to see how EMCICs af- resenta conservativeestimate ofblast-wavetemperatureand
fect these parameters. In Figure 5, the p distributions for flowstrengthstotheparentdistributions.
T
p+pcollisions and the six lowest multiplicity selectionson In summary, to the extent that the curves in Figure 3 de-
Au+Aucollisionsareshown.Blastwavefitstothemeasured scribetheratiosshownthere–whichtheydoin sign, magni-
spectra,resultingintheparametersshownbyredtrianglesin tudeandmassdependence,butonlyapproximatelyinshape–
Figure2areshownascurves. OnthelinearscaleoftheFig- the data is consistent with a common parent distribution for
ure, some deviationsbetweenthe fit anddata, particularlyat spectrafromallcollisions.TheresidualdeviationseeninFig-
the lowest p for the light particle, is seen. This has been ure3isobservedagainindifferentformsinFigures4and2.
T
observedpreviouslyinBlast-wavefits,andmaybeduetores- The upshot is that EMCICs may dominate the multiplicity
onances[19, 46]. Nevertheless,thefits to measureddataare evolutionofthespectrainthesoftsectoratRHIC.Extracting
reasonableoverall,andforsimplicity,wedonotexcludethese physics messages from the changing spectra, while ignoring
bins. kinematic effects of the same order as the observed changes
Also shown in Figure 5 are the “EMCIC corrected”spec- themselves,seemsunjustified.
tra, as discussed above. As already seen in Figure 4, these In particular, STAR [21] and others [19] have fitted the
differ fromthe measured spectra mostly for low multiplicity spectra with Blast-wave distributions, which ignore EMCIC
collisions and for the heavier emitted particles. Blast-wave effects.Basedonthesefits,theyconcludedthatthedifference
fits to these spectra are also shown. Especially for the very in spectral shapes between high- and low-multiplicity colli-
lowest multiplicity collisions, these fits are less satisfactory sionswasduetomuchlowerflowinthelatter;c.f. Figure2.
thanthosetothemeasuredspectra;the“parentdistributions” Recently,Tangetal.[47]arrivedtothesameconclusion,us-
8
2 p+p 0.1 0.06
0.04
1 p - 0.05 K- 0.02 p
15 Au+Au [70-80]% 0.4
] 1
2V 10 0.2
Ge 5 0.5
2/
c 30
y) [ 20 Au+Au [60-70]% 2 00..46
dT 10 1 0.2
p
d
T 4 1
p Au+Au [50-60]%
40
p2 20 2 0.5
(
N/
2d 100 1.5
Au+Au [40-50]% 6
1
50 4
2 0.5
Au+Au [30-40]% 10 2
100
5 1
50
3
150 Au+Au [20-30]% 10
2
100
5
50 1
0.2 0.4 0.6 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.4 0.6 0.8 1
p [GeV/c] p [GeV/c] p [GeV/c]
T T T
FIG. 5: (Color online) dN/dp2 spectra for pions (left), kaons (center), and protons (right) are plotted on a linear scale, as a function of
T
event multiplicity. Toppanelsshowspectraforminimum-bias p+pcollisions, andthespectraforthesixlowestmultiplicityselectionsof
Au+Au collisions are shown inthe lower panels. Filledsymbols are the measured data, while open symbols are the “EMCIC corrected”
distributions,discussedinthetext.(Forpions,thesedistributionsoverlapalmostcompletely.) Blast-wavefitsareindicatedbythecurves. For
the“EMCICcorrected”spectra,twofitsareperformed,toestimatesystematicerrors.Thesolidlinerepresentsafittoalldatapoints,whilethe
fitindicatedbythedashedlineignoresprotonyieldsabovepT =0.8GeV/c.
ingamodifiedBlast-wavefitbasedonTsallisstatistics. This B. Softsector:unidentifiedparticlesinmultiplicity-selected
requiresintroductionofanextraparameter,q,intendedtoac- p+pcollisions
countfor system fluctuationeffects [48]. However, contrary
totheclaimsintheTangpaper,theTsallisdistribution-with
While minimum-bias p+pcollisionsare the natural“ref-
or without q - does not account for energy and momentum
erence”whenstudyingAu+Aucollisions, the STARexperi-
conservation[49];EMCICeffectswouldneedtobeaddedon
menthasalsomeasuredp spectrafrommultiplicity-selected
T
the top of the Tsallis statistics [49]. Therefore, conclusions
p+p collisions [23]. These are reproduced in Figure 6, in
about flow in low-multiplicity collisions based on these fits
whichthelowest-multiplicitycollisionsareshownonthebot-
aresuspect.
tom andthe highestatthe top. Numericallabelsto the right
ofthespectraareincludedjustforeaseofreferencehere.
Thesolidcurveisapower-lawfittothehighest-multiplicity
spectrum (#10), just for reference. This curve is scaled and
replottedasdashed lines, to make clear the multiplicityevo-
lution of the spectra. Concentrating on the soft sector for
An independent measurement of flow would help clarify the moment, we perform the same exercise as above, to see
this issue. Two-particle femtoscopy (“HBT”) is a sensitive towhatextentthismultiplicityevolutioncanbeattributedto
probe of collective motion [50] and has been measured in EMCICs.
p+pcollisionsatRHIC[51]. Anyscenarioshouldbeableto InFigure7areshownthreeratiosofspectra,inwhichthe
describesimultaneouslyboth the spectralshapesand the m second-highest-multiplicityspectrum (#9) is used as the de-
T
dependenceofthefemtoscopicscales. Astudyofthistopicis nominator,toavoidstatisticalfluctuationsassociatedwiththe
underway. highestmultiplicity spectrum. Also shown are curves, using
9
1016
1015 1.2
1013
-2]) 1011 )max 1
c N
eV/ 109 ,pT0.8
G 9 (c
[(pT107 )/fNi0.6 #1 / #9
dn/d 105 7 ,(pcT0.4 #4 / #9
) T103 f0.2
p #7 / #9
1/ 10 5
) (h 0
nc10-1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
(1/ 10-3 3 pT [GeV/c]
10-5 FIG.7: (Coloronline)Ratioofthe pT spectrashownbyfullpoints
1 inFigure6. Spectraforthelowest-multiplicity(redtriangles),fifth-
10-70 1 2 3 4 5 6 lowest (green triangles) and seventh-lowest (squares) multiplicity
p [GeV/c] collisions are divided by the spectrum for the second-highest mul-
T tiplicitycollisions. Curvesrepresentacalculationofthisratio(ratio
ofEMCICfactors)usingEquation6;seetextfordetails.
FIG.6:(Coloronline)Transversemomentumspectraofunidentified
negativehadronsfrom p+pcollisionsat√sNN =200GeVbythe
STARCollaboration[23]. Thelowest(highest)datasetcorresponds
to the lowest (highest) multiplicity collisions. Thesolid line isin- C. Segue:Fromthesofttothehardsector
tended only to guide the eye and show the shape of the spectrum
for thehighest multiplicityselection. Itisrescaled andredrawnas Figure3showsthecentralresultofthispaper:namely,that
dashed lines below, to emphasize the multiplicity evolution of the
the multiplicityevolutionof the mass and p dependenceof
spectrumshape. T
single particle spectra in the soft sector may be understood
almost entirely in terms of phase-space restriction with de-
creasingeventmultiplicity.
Equation 6 with the energy-momentum scales given in Ta-
Plotted in that figure is the ratio of spectra from low-
bleV.
multiplicityeventsoverspectrafromhigh-multiplicityevents.
The spectra reported by STAR are for unidentified nega-
Experimentalstudiessometimesshowthisratio’sinverse,of-
tive hadrons. In calculating these curves, we assumed that
ten called R [3]. While of course the same informationis
AA
allparticleswerepions. Thismatters,sincetheenergyterms
showninbothrepresentations,wechoosethatofFigure3for
in Equation 6 require the particle mass. We expect the
tworeasons. ThefirstistoemphasizetheeffectsofEMCICs,
energy-momentumscales listed in Table V to be affectedby
thetopicofthispaper;theseare,generically,tosuppressthe
this simplistic assumption. Particle-identified spectra from
particleyieldathighenergyandmomentum,particularlyfor
multiplicity-selectedp+pcollisionswouldberequired,todo low-N final states. (In multiparticle distributions, they also
better. Given this, and the only semi-quantitative agreement
generatemeasurablecorrelations[25].)
between the calculations and measured ratios shown in Fig-
Thesecondreasonistostressthatwehavebeendiscussing
ure 7, we conclude only that the EMCIC contributionto the
spectra in the soft sector, whereas the ratio R is generally
AA
multiplicityevolutionoflow-p spectrain p+pcollisionsis
T studied at high p . At large p , we expect that a purely
T T
atleastofthesameorderastheobservedeffectitself.
EMCIC-basedexplanationofthemultiplicityevolutionofthe
spectra might break down, for two reasons. Firstly, even if
particlesof allmomentasharedphase-spacestatistically, our
Multiplicitycut N p2 [(GeV/c)2] E2 [GeV2] E [GeV]
h Ti h i h i approximation of Equation 2 is expected to break down for
#1 6.7 0.31 0.90 0.84 energiesmuchabovetheaverageenergy,asdiscussedinAp-
#4 11.1 ” ” ” pendixB. Secondly,itisbelievedthatthehigh-pT yieldhas
#7 24.2 ” ” ” a large pre-equilibrium component; thus, high-pT particles
mightparticipatelessinthestatisticalsharingofphase-space,
#9 35.1 ” ” ”
asdiscussedinSectionIIC.
TABLE V: Multiplicity and parent-distribution kinematic parame- As we discuss in the next Section, EMCICs surely do
ters which give a reasonable description of the spectrum ratios for not dominate the multiplicity evolution of the hard sector in
unidentified particles in the soft sector from multiplicity-selected heavy ion collisions. For interpreting high-p spectra from
T
p+p collisions. See text for details. Note that the multiplicity multiplicity-selectedp+pcollisions,accountingforEMCICs
changeswitheventclass;theparentdistributionisassumedidentical.
mayormaynotbeimportant.Inordertomaketheconnection
10
1.2
#9 / #1
) 1
x
a #9 / #4
m 10
N
,pT0.8 ) T #9 / #7
( p
)/fNci0.6 (Rpp
,pT0.4
(
c
f
0.2
1
0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
p [GeV/c] p [GeV/c]
T T
FIG.8: (Color online) Thesamedataandcurves asshown inFig- FIG.9: (Coloronline)“Rpp,”theanalogueof“RCP”usedinheavy
ure7,butplottedovertheentiremeasured pT range. ioncollisions.Thespectrumfromthehighest-multiplicityp+pcol-
lisionsaredividedbyspectrafromlower-multiplicitycollisions(see
filled datapoints in Figure 6). The data and curves are simply the
to Figure 3, we will plotspectra from low-multiplicitycolli- inverseofthoseshowninFigure8.
sionsoverthosefromhigh-multiplicity,aswellastheinverse,
tomaketheconnectiontoR .
AA
calculationsandmeasurementsleadsustoconcludethatEM-
CICs do not fully explain the multiplicity evolution of p
T
spectrain p+pcollisionsinthehardsector.
D. Spectrainthehardsector
However,this,initself,raisesafascinatingpossibility.Fig-
ure8showsthat,relativetohigh-multiplicityp+pcollisions,
ThegenericeffectofEMCICsistosuppressparticleyields
the suppressionof high-p yieldsfrom low-multiplicitycol-
T
at energy-momentumfar from the averagevalue. The effect lisionsisnotasstrongasoneexpectsfromoursimplepostu-
isstrongerforlowermultiplicityN. Itisclear,then,thatEM-
late. Said anotherway, the high-p “enhancement”in high-
T
CICscannotaccountforthemultiplicityevolutionofthespec- multiplicitycollisionsmaynotbeaslargeasoneexpectsfrom
tra at high pT in Au+Au collisions, since high-multiplicity phasespaceconsiderationsalone. Thisis emphasizedin Fig-
collisions are observed to have more suppression at high pT ure9,inwhichisplotted“Rpp”,theratioofthespectrumfrom
than do low-multiplicity collisions [3]. Thus, we conclude high-multiplicity to lower-multiplicity collisions; R is the
pp
thatourpostulatefailsfor Au+Aucollisionsathigh pT; the analogofRCPfromheavyioncollisions[3].
“parentdistribution”describingtheunderlyingphysicsinthis The motivation for studying quantities like R and R
AA CP
regiondoes,indeed,changewithmultiplicity. (and now R ) is to identify important differences between
pp
But in p+p collisions, the multiplicity evolution in the oneclassofcollisionsandanother. Presumably,oneisinter-
hardsectorisoppositetothatinAu+Aucollisions. Inpartic- ested in physics effects (jet quenching, etc.), above and be-
ular,in p+pcollisions,theyieldathighpT (relativetolower yond “trivial” energyand momentumconservation. Thus, it
pT)isincreasedasmultiplicityincreases,asisclearfromFig- makes sense to attempt to “correct” for EMCICs by divid-
ure 6; similar results have been observedin p+p collisions ingthemoutas wedid inSectionIIIA, keepingin mindthe
attheTevatron[52],ISR[53],andSppS[54]. A“hardening” caveatsjustdiscussed.
ofthespectrumwithincreasingmultiplicitygoesinthesame TheresultofthisexerciseisshowninFigure10, inwhich
directionas would EMCIC effects. To what extentcan EM- the datapoints from Figure 9 are divided by the curves from
CICs account for the multiplicity evolution of spectra from thesameFigure,toformanewquantity,R . Explicitly,the
′pp
p+pcollisions,inthehardsector? greencirclesonFigure10,whichcomparemultiplicityselec-
SomeinsightonthisquestionmaybegainedfromFigure8, tions#9and#4aregivenby
inwhichthedataandcurvesshowninFigure7areplottedout
to pT =6GeV/c. Clearly,thecalculatedsuppressionfunction dn
(EqWueatiroenca6ll)ftahialst dErqaumaatitoicnasll2yaatnhdig6haprTe.based on the cen- R′p(#p9,#4)(pT)≡ ddpnT(cid:12)(cid:12)#9× (7)
tral limit theorem (CLT), which naturally leads to Gaussian dpT(cid:12)#4
(cid:12)
distributions. As discussed in Appendix B, one expects the 1 (cid:12) 1 2p2 (E E )2
exp (cid:12) T + −h i ,
bdrisetarkibduotwionn–ofe.tgh.ewChLeTn pap2proxipm2at.ioTnhuins,tahneyfainrftearielnscoefstwhee "(cid:18)2(N#9−1)−2(N#4−1)(cid:19) hp2Ti hE2i−hEi2!#
T ≫h Ti
make about EMCIC effects in the hard sector remain quali- where the relevantquantitiesfromTable V are used. Again,
tative. Nevertheless, the level of disagreement between the all particles are assumed to have pion mass. Qualitative