Table Of ContentA Fluctuation Probe of Disoriented Chiral Condensates
B. Mohanty,1 D.P. Mahapatra1 and T.K. Nayak2
1Institute of Physics, Bhubaneswar-751005, India and
2Variable Energy Cyclotron Centre, Kolkata-700064, India
(Dated: February 8, 2008)
We show that an event-by-eventfluctuation of the ratio of neutral pions or resulting photons to
charged pionscan beusedas an effectiveprobefor theformation of disoriented chiralcondensates.
Thefact that theneutralpion fraction produced in case of disoriented chiral condensate formation
has a characteristic extended non gaussian shape, is shown to be the key factor which forms the
basis of thepresent analysis.
PACSnumbers: 25.75.+r,13.40.-f,24.90.+p
2
0 Apart from studying the dynamics of nuclear matter ing distribution for non-DCC events is a gaussian with
0 at extreme conditions of temperature and density, the < f >= 1/3. It can easily be seen that for events with
2
main goals of relativistic heavy-ioncollision experiments DCC there is a strong anti-correlationin the production
n have been to probe for the possible existence of quark- ofneutraltochargedpions. Thisisthekeyfeaturewhich
a gluon plasma (QGP) and chiral symmetry restoration. forms the basis of the present analysis.
J
It is also believed that the hot and dense nuclear matter Tostartwith,wedefinetheratio,RasN /N ,where
π0 ch
5 formed in the collision, while going through the expan- N is the number of neutral pions and N = N +
π0 ch π+
2
sion and cooling stage, may relax into a vacuum state Nπ−,isthemultiplicityofchargedpions. Thefluctuation
oriented quite differently from the normal ground state. in the ratio R is given as
2
v This results in the formation of what are known as Dis-
0 oriented Chiral Condensates (DCC) which finally decay
1 producing an imbalance in pion production. In such a D = <δR2 > = <δNπ20 > + <δNc2h >
0 casetheprobabilitydistributionofthe neutralpionfrac- <R>2 <N >2 <N >2
π0 ch
1
tion is very different from what one would expect in the <δN δN >
0 2 ch π0 (2)
absence of any DCC production [1, 2]. It has also been
2 − <Nch ><Nπ0 >
suggested that DCC formation can be taken as a signal
0
/ of chiral symmetry restoration at high temperature and where the average (< >) is over events with
x density[3,4]. TheoreticalcalculationsonlifetimeofDCC <δR2 >=<R2 > <··R· >2.
e −
- givesencouragingresultsonitsdetectioninactualexper- Sincethe totalprobabilityis always1,knowingNπ0 =
l iments[5]. Severalexperiments,startingfromcosmicray f N , one canwrite N =(1 f)N . Followingsimple
c π ch π
−
u studies[6],hadron-hadroncollision[7],tothepresentday statistical analysis, we can show
n heavy-ion collision experiments [8, 9] in which attempts
: havebeen made to look for DCC signal, haveeither pro-
v
δN δN δf
i ducednullresultsorhavemanagedtoputanupperlimit π0 = π + (3)
X in DCC production. <Nπ0 > <Nπ > <f >
r Recently fluctuations in the ratio of positive to nega-
a and
tivechargeandbaryontoanti-baryonnumbershavebeen
shown to be effective probes for the possible existence
δN δN δf
ch π
of QGP [10, 11, 12]. Fluctuations in baryon number = (4)
<N > <N > − (1 <f >)
or charge asymmetry have been shown to be very much ch π −
dependent on whether the system goes through a QGP
By squaring Eq. (3) and Eq. (4) one can get the
phase or a purely hadronic gas phase. In the present contributions to the terms <δN2 >/<N >2 and
letter we show that a similar event-by-event fluctuation <δN2 >/<N >2 in Eq. (2). Wπ0hile by mπ0ultiplying
study of the neutral to charged pion ratio can be an ef- ch ch
Eq. (3) and Eq. (4), the contribution to the cross term
fective probe for DCC production.
<δN δN >/<N ><N > in Eq.( 2) is obtained.
ThebasicdifferencebetweentheeventswithDCCand ch π0 ch π0
So the fluctuation in ratio R = N /N , as given by
those without any DCC formationlies in the probability π0 ch
Eq. (2) becomes
distributionofthe neutralpionfraction. Foreventswith
DCCtheprobabilitydistributionofneutralpionfraction,
f, is given by <δf2 >
D = (5)
2 2
P(f)=1/2 f where f =N /N , (1) <f > (1 <f >)
π0 π −
p
N andN being the totalnumberofneutralpions and This shows that the fluctuation in the ratio, R, is
π0 π
the total number of pions respectively. The correspond- essentially dominated by the fluctuation in the neutral
2
pion fraction whose probability distribution for DCC
is given in Eq. (1). Now let us see, what is the value
of D = <δR2 >/<R>2 for non-DCC and DCC cases
respectively.
1
Non-DCC case. - Itisknownfromppexperiments [13]
that the produced pions have their charge states parti-
tionedbinomiallywiththemeanoff at1/3. Thefluctu- D)
ation in f is inversely proportional to the total number n (
as given by <δf2 > ∼ 2/9N1π. Then one can write uctuaio10-1 Np = 300
Dnon−DCC = N <f >2(1 <f >)2 ∼4.5/Nπ (6) Fl
π
−
DCC case. - For the DCC case, the probability distri-
bution is given by Eq. (1). Using this, one can easily
see that < f >= 1/3, as in the non-DCC case, while
<δf2 >=4/45. Substituting these values in Eq. (5), we 10-2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
find, DCC event fraction (a )
D =1.8 (7)
DCC
For a typical case, where the total pion multiplicity, N ,
π FIG. 1: Variation of fluctuation (D) with the fraction of
in the experiment is 300, we find for the non-DCC case
DCC events(α). The plot is obtained for a average detected
the above fluctuation is only about 0.015. This is way
pion multiplicity of 300.
below the value 1.8 obtained for DCC.
In a typical reaction DCC may or may not be formed
andifformedtherecouldevenbebackgroundpionscom- value π/2 if f < 1/2 and π/2 2arccos(1/2 (f)) for
−
ingfromnon-DCC sources. Therecouldalsobe multiple f >1/2. For higher number of domains one haps to com-
DCC domains formed. The effects of these are discussed pute it numerically. It can be shown that the resultant
below. probability distribution approaches a gaussian centered
Ensemble of DCC and non-DCC events. - In reality, at1/3withthestandarddeviation 1/√m. Thismeans
2 2 ∼
DCC formation depends on several factors [14]. That <δf >/<f > 1/m, so that the fluctuation, D, re-
∼
meansnotalleventsinacollisionarefavourableforDCC duces as the number of domains increases in an event.
production. If the sample of events is a mixture of DCC However, by carrying out this analysis by dividing the
and non-DCC events, the signal should depend on the η φ phase space to appropriate bins this effect can be
−
respective fractions. To a first approximation, the cor- reduced.
responding fluctuation would be a linear combination of Events with pions from DCC as well as non-DCC
results shown in Eq. (6) and (7) as given by sources. - Inagiveneventpionscanoriginateboth from
a DCC source and from other non-DCC sources even in
D =αDDCC +(1−α)Dnon−DCC (8) a DCC event. In such a case the probability distribu-
where α is the DCC event fraction. Fig. 1 shows the tionfortheneutralpionfraction,P(f),canbeevaluated
variation of fluctuation, D, with the fraction of DCC following Eq. (9) as
events (α), obtained from Eq. (8), for a typical case of
average total pion multiplicity of 300.
Multiple DCC domains. - The fluctuation of DDCC as P(f)=Z dfDCC dfnon−DCC
given in Eq. (8) is valid for pions emitted from a single
region or domain of DCC. It is possible that in an event P(fDCC) P(fnon−DCC)
multiple domains of DCC are formed. In sucha case the δ(f βfDCC (1 β)fnon−DCC) (10)
totalprobabilitydistributionofthe neutralpionfraction − − −
istheaveragevalueofP(f)overallofthedomains. This where β corresponds to fraction of DCC pions out of a
can be written as total number of pions N , produced in the event. f
π DCC
f1+ +fm and fnon−DCC correspond to the neutral pion fraction
Pm(f)=Z df1···dfmδ(f − ·m·· ) for pions from DCC and non-DCC sources in the event,
respectively. With the presence of non-DCC pions, one
P1(f1)···Pm(fm) (9) can see, the neutral pion fraction can no longer start
where, m is the number of domains and for the non- fromzero nor canreachthe full value of unity, the range
DCC case P(f) δ(f 1/3). For a case with two do- depending upon the fraction of non-DCC pions present
∼ −
mains Eq. (9) can be evaluated analytically to yield a in the sample. For various values ofβ, andtotalnumber
3
<δǫ2 > <δǫ2 >
Dexp =D+ 1 + 2 (12)
<ǫ1 >2 <ǫ2 >2
1
providedweassume thatthe efficiency ofdetectionis in-
dependentofmultiplicity. Thisisnotabadassumptionif
theanalysisiscarriedoutoveranarrowbinincentrality.
D) Withdetectorsformeasuringphotonandchargedpar-
on ( ticlemultiplicityinacommoncoverageoftheη φphase
ai −
u -1 spaceasin WA98[8]inCERNSPS,STAR inRHIC and
uct10 Np = 300 ALICE at LHC, the present method of analysis is ex-
Fl
pectedtobeveryusefulregardingthesearchforDCCsig-
nal. The observed photon multiplicity, Nγ−like, in such
a situation is usually contaminated with some charged
particles that mimic photons. However, the true pho-
tonmultiplicity, N canbe extractedfromthe measured
γ
-2
10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Nγ−like valueaftercorrectingfortheefficiencyofphoton
DCC pion fraction (b ) counting and the purity of photon sample. This is done
using the relation [15],
p
γ
FIG.2: Variationoffluctuation(D)asafunctionofthepion Nγ = Nγ−like (13)
ǫ
γ
fraction (β) from a DCC origin. The plot is obtained for an
average pion multiplicity of 300. where, ǫ is the photon counting efficiency and p is the
γ γ
purity of the photon sample obtainedfromdetector sim-
ulations. This will modify the expression for fluctuation
given in Eq. (12). If we assume that the purity and effi-
ofnon-DCCpions,(1 β)Nπ,theaboveequationcanbe ciencyarenotdependentonmultiplicityandindependent
−
evaluated easily to obtain the final f distribution from of each other, we get
which one can get the resultant fluctuation, D, using
Eq.(5). Theresultsobtainedforanaveragetotalnumber <δǫ2 > <δǫ2 > <δp2 >
Dexp =D+ γ + 2 + γ (14)
of 300 pions in an event, are shown in Fig. 2. One can <ǫγ >2 <ǫ2 >2 <pγ >2
seethatthefluctuationD,decreaseswithdecreaseinthe
fraction of DCC pions. Sensitivity to DCC signal. - Let us consider an exper-
Neutral pion decay. -Neutral pions are not actually de- iment where the average total detected pion multiplic-
tected. They are rather reconstructed from photons de- ity is 300. Taking typical values of relative fluctuation
tected using electromagnetic calorimeters. Basically one (σ/mean)inefficiency andpurity,whichareofthe same
detects the decay photons (π0 2γ). For such a case order, to be 3% [16], we have <δǫ2 >/<ǫ>2 and
one can take Nπ0 = 2Nγ. Ho−w→ever we find that this <δp2γ >/<pγ∼>2 both to be about 0.0009. Together
substitution leaves the expression in Eq. (2) unchanged. they introduce an error 0.0027 in the value of fluctua-
That is tion, Dnon−DCC, which now has a value 0.015 0.0027.
±
From Fig. 1 we find that corresponding to D 0.02
∼
which is the case for pions from non-DCC events for
D =D (11)
γ a typical experiment as described here, the value of
α = 0.002. This means that the analysis is sensitive if
where Dγ corresponds to the fluctuation in the ratio the fractionofDCC events ina givensample ofeventsis
Nγ/Nch. That makes this observable actually measur- above0.2%,providedallthe pionsinthesampleofDCC
able, event-by-event, in an experiment and the con- events are of DCC origin. From Fig. 2 we find a value
clusions derived above for the Nπ0/Nch hold true for of β = 0.025 corresponding to D 0.02. This means
Nγ/Nch. In an actual experiment, such as in STAR at that in a sample of DCC events, th∼e method is sensitive
RHIC, one can measure Nγ and Nch and look for the if more than 2.5% of the pions are of DCC origin.
signal of DCC using this method. Combining the results from Fig. 1 and Fig. 2 one can
We know that the measurement of any observable in obtain a sensitivity curve for such a typical experiment.
anexperimentisaffectedbytheefficiencyofthedetector. This is shown in Fig. 3. It shows, the sensitivity of the
Theeffectofthiscanbeseeneasilythroughthefollowing method in a typical experiment to the fraction of DCC
calculation. Consider the efficiency of detecting photons pions, β, in an event vs. the fraction of DCC events,
is ǫ1 and that for charged pions is ǫ2. Then we have, α. The contour is drawn corresponding to a value of
Nγexp = ǫ1Nγ and Ncehxp = ǫ2Nch. Following the simple D ∼ 0.02. The excluded region, which is to the left of
statisticalanalysisasusedforarrivingatEq.(3),weget, the curve, indicates the region which cannot be probed
4
RHIC and LHC, where the number of emitted pions are
significantlylarge,the fluctuationinthe ratioofphotons
1
to charged particles for non-DCC sources are going to
be quite small. Thus the present method will be quite
sensitive for detecting signals of DCC at these energies.
The effect of acceptance on ratio fluctuations such as
)
an (10-1 Np = 300 ours, have been discussed in Ref. [10]. As has been ar-
o gued there, the results are expected to be independent
acti of acceptance. Some more generaldiscussionon effect of
nt fr acceptance on multiplicity fluctuations can be found in
e
v Ref [17].
e
CC 10-2 In summary, we have shown that detection of DCC is
D possible through an event-by-event fluctuation study of
Excluded Region
theratioofmultiplicitiesofneutraltochargedpions. For
acase,withDCCproduction,theabovefluctuationhasa
valuecloseto2whileforanon-DCCcasethesameturns
10-3 outtobemuchsmaller,inverselyproportionaltothepion
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
DCC pion fraction (b ) multiplicity. Howeverinanactualexperiment, event-by-
event measurement of neutral pion multiplicity may not
be feasible. Instead one detects photons coming from
its decay. We have shown that the conclusions drawn
FIG. 3: The sensitivity of a typical experiment in terms
from the fluctuation in N /N remain unchanged for
of the fraction of DCC type events, α, for a given fraction π0 ch
the fluctuation in the ratio of photon and charged pion
of pions coming from DCC origin, β, corresponding to an
multiplicities(N /N ). Theanalysishasbeenextended
average detected pion multiplicity of 300. γ ch
to include pions from both DCC and non-DCC sources
in the same event. We have discussed other detector re-
latedeffects onthisanddemonstratedthattheproposed
by such an experiment. For example, the experiment is fluctuationanalysiscaneasilybeappliedtoexperimental
sensitive to DCC with 20% of the pions coming from a data regarding DCC search.
DCC origin, provided there are more than 4 events in The authors wish to thank S. C. Phatak, Institute of
every 100 events which are DCC-type. By improving on Physics,andN.Barik,DepartmentofPhysics,UtkalUni-
thephotondetectionandreducingthefluctuationineffi- versity, Bhubaneswar for some very helpful suggestions
ciency and purity of measuredphoton sample, the sensi- and discussions.
tivity ofthe methodcanbe improved. Inexperiments at
[1] J.D. Bjorken, Int. J. Mod. Phys. A7, 4189 (1992). [10] S. Jeon, V.Koch, Phys. Rev.Lett. 85, 2076 (2000).
[2] J. -P. Blaizot and A. Krzywicki, Phys. Rev. D46, 246 [11] M. Asakawa, U. Heinz, B. Muller, Phys. Rev. Lett. 85,
(1992). 2072 (2000).
[3] K. Rajagopal and F. Wilczek, Nucl. Phys. B399, 395 [12] S.A.Bass,P.Danielewicz, S.Pratt,Phys.Rev.Lett.85,
(1993); Nucl. Phys.B404, 577 (1993). 2689 (2000).
[4] S.Gavin,A.GockschandR.D.Pisarski,Phys.Rev.Lett. [13] P. Grassberger, H.I. Miettinen, Nucl. Phys. B89, 109
72, 2143 (1994). (1975).
[5] J.V. Steele and V. Koch, Phys. Rev. Lett. 81, 4096 [14] A. Krzywicki and J. Serreau, Phys. Lett. B448, 257
(1998). (1999).
[6] C.R.A. Augustoet al., Phys.Rev. D59, 054001 (1999). [15] M.M.Aggarwaletal.,(WA98Collaboration),Phys.Lett.
[7] T.C.Brooksetal.,(MinimaxCollaboration), Phys.Rev. B458, 422 (1999).
D61, 032003 (2000). [16] M.M.Aggarwaletal.,(WA98Collaboration),Submitted
[8] M.M.Aggarwaletal.,(WA98Collaboration),Phys.Lett. to Physical ReviewC, eprint: nucl/ex-0108029.
B420, 169 (1998); Phys. Rev.C58, 011901(R) (2001). [17] D.P.Mahapatra, B.Mohanty,S.C.Phatak,e-print:nucl-
[9] H. Appelshauser et al., (NA49 Collaboration), Phys. ex/0108011; To be published in Int. J. Mod. Phys. A.
Lett.B459, 679 (1999).
