Table Of ContentSqueezed K+K− correlations in high energy heavy ion collisions
Danuce M. Dudek and Sandra S. Padula∗
Instituto de F´ısica Te´orica–UNESP, C. P. 70532-2, 01156-970 S˜ao Paulo, SP, Brazil
(Dated: January 5, 2011)
Thehotanddensemediumformedinhighenergyheavyioncollisionsmaymodifysomehadronic
properties. In particular, if hadron masses are shifted in-medium, it was demonstrated that this
couldleadtoback-to-backsqueezedcorrelations(BBC)ofparticle-antiparticlepairs. Althoughwell-
established theoretically, the squeezed correlations have not yet been discovered experimentally. A
methodhasbeensuggestedfortheempiricalsearchofthiseffect,whichwaspreviouslyillustratedfor
φφpairs. Weapplyheretheformalism andthesuggestedmethodtothecaseofK+K− pairs,since
they may be easier to identify experimentally. The time distribution of the emission process plays
acrucial role in thesurvivalof theBBC’s. Weanalyze thecases where theemission issupposed to
occur suddenly or via a Lorentzian distribution, and compare with the case of a L´evy distribution
1
in time. Effects of squeezing on thecorrelation function of identical particles are also analyzed.
1
0
2
DOI:10.1103/PhysRevC.82.034905 described by analogous formalisms, being both positive
n correlations with unlimited intensity. This last feature
a
contrasts with the observed quantum statistical correla-
J
tions of identical bosons and identical fermions, whose
3 INTRODUCTION
intensities are limited to vary between 1 and 2, or 0 and
] 1, respectively. In the remainder of this paper, we focus
h Since the beginning of the 1990’s,some people started
our discussion on the bosonic case only.
t calling attention to the possible existence of a different
- The correlation reflecting the squeezing is quantified
l typeofcorrelation,occurringbetweenparticlesandtheir
c in terms of the ratio of the two-particle distribution by
u antiparticles. Initially, in 1991, Weiner et al.[1] pointed
the product of the single-inclusive distributions, i.e., the
n out to the surprise existence of a new quantum statis-
spectra of the particle and of the antiparticle. For the
[ tical correlation between π+π−, which would be simi-
sake of comprehension, we first briefly discuss the for-
lar to the π0π0 case (since π0 is its own antiparticle),
2 malism for bosons that are their own antiparticles, such
v butentirelydifferentfromthe Bose-Einsteincorrelations
as φφ or π0π0 pairs. In this case, the full correlation
9 (betweenπ±π±)leadingtothe Hanbury-Brown&Twiss
function, after applying a generalization of Wick’s the-
9 (HBT) effect. They related those correlations to the ex-
8 orem for locally equilibrated systems [5, 6] consist of a
pectation values of the annihilation (creator) operators,
5 part reflecting the identity of the particles (HBT), and
< aˆ(†)(k )aˆ(†)(k ) >= 0, which was then estimated by
6. 1 2 6 another one, reflecting the particle-antiparticle squeezed
usingadensitymatrixcontainingsqueezedstates,analo-
0 correlation(BBC). This can be written as
gous to two-particle squeezing in optics. They predicted
0
1 that such squeezed correlations would have intensities
N (k ,k )
: above unity, either for charged or neutral pions, i.e., C (k ,k )= 2 1 2
iv Cs(π+π−) > 1 and Cs(π0π0) > 1. Later, Sinyukov[2], 2 1 2 N1(k1)N1(k2)
X discussedasimilareffectforπ+π− andπ0π0 pairs,claim- G (1,2)2 G (1,2)2
c s
=1+ | | + | | . (1)
r ing that they would be due to inhomogeneities in the G (1,1)G (2,2) G (1,1)G (2,2)
a c c c c
system, in opposition to homogeneity regions in HBT,
Theinvariantsingle-particleandtwo-particlemomentum
comingfromahydrodynamicaldescriptionofthesystem
distributions are given by
evolution.
moOrethaecrctuernattaetliyv,eamndodietlsfitnraielldythoafpoprmenueldateatthtehepreonbdlemof Gc(i,i) = ωkihaˆ†kiaˆkii=ωkidd3kN
i
t[3h]a.tIdnecthadeier,ainpparpoarochp,ostihteiosne msqaudeeezbeydMba.cAk-staok-abwaacketcoarl-. Gc(1,2) = √ωk1ωk2haˆ†k1aˆk2i,
relations(BBC)ofboson-antibosonpairsresultedfroma Gs(1,2) = √ωk1ωk2haˆk1aˆk2i. (2)
quantum mechanical unitary transformation relating in- Inthe aboveequations, ... representsthermalaverages.
h i
medium quasi-particles to two-mode squeezed states of The first term in Eq. (2) corresponds to the spectrum
their free counterparts. We discuss it in some more de- of each particle, the second is due to the indinstinguibil-
tail below. Shortly after that, P. K. Panda et al.[4] pre- ity of identical particles, reflecting their quantum statis-
dicted that a similar BBC between fermion-antifermion tics. The third term, in the absence of in-medium mass
pairs should exist, if the masses of these particles were shift is in general identically zero. However, if the parti-
modifiedin-medium. Boththefermionic(fBBC)andthe cle’smassismodifiedin-medium,itcancontributesignif-
bosonic (bBBC) back-to-back squeezed correlations are icantly, triggering this novel type of particle-antiparticle
2
correlation, yet to be discovered experimentally. This is RESULTS FOR K+K− PAIRS
achieved by means of a Bogoliubov-Valatin (BV) trans-
formation, which relates the asymptotic creation (anni-
Initial studies of the problem were performed for a
hilation) operators,aˆ†k (aˆk), ofthe observedbosons with static, infinite medium [3, 4], later extended to a finite-
momentum kµ=(ωk,k), to the in-medium operators,ˆb†k size system, radially expanding with moderate flow[7].
(ˆbk), corresponding to thermalized quasi-particles. The For simplicity, a non-relativistic approach was consid-
BV transformation is given by ered, assuming flow-independent squeezing parameter.
The expansion of the system was described by the emis-
aˆk =ckˆbk+s∗−kˆb†−k ; aˆ†k =c∗kˆb†k+s−kˆb−k , (3) sion function from the non-relativistic hydrodynamical
parameterization of Ref.[9], later shown to be a non-
being c =cosh(f ) and s = sinh(f ); ( k) denotes an
k k k k − relativistic hydrodynamical solution. In Fig. 1 we illus-
oppositesigninthe spacialcomponentsofthe momenta.
trate these assumptions with a simple sketch. The flow
For conciseness, we keep here the short-hand notation
velocity during the system expansion was considered as
introduced in Ref.[3] . The coefficient
v = u r/R. The values u = 0 and u = 0.5 are used
h i h i h i
in the present work. Within the hypotheses described
f (x)= 1log Kiµ,j(x)uµ(x) , (4) above, analytical results were obtained for the squeezed
i,j 2 K∗ν(x)u (x)
" i,j ν #
is the squeezing parameter, where Kµ (x) = 1(kµ+kµ)
i,j 2 i j
is the averageofthe momenta of eachparticle, andu is
µ
the flow velocity of the system. The BV transformation
between the operators is equivalent to a squeezing oper-
ation, from which the name of the resulting correlation
is derived.
Incaseofchargedmesons,suchasπ±orK±,theterms
in Eq. (1) would act independently, i.e., either the first
and the second terms together would lead to the HBT
effect (for π±π± or K±K± pairs), and the first and the
lastterms,totheBBCeffect(forπ+π− orK+K− pairs).
The in-medium modified mass, m , was originally[3]
∗
related quadratically to the asymptotic mass, m, i.e.,
m2(k)=m2 δM2(k),wheretheshiftinginthemass, FIG. 1: Sketch illustrating the cross-sectional area of the
δM∗(|k|), could−depen|d|on the momenta of the particles. Gaussian profile (∼e−r/(2R)2) of the system expanding with
| |
Nevertheless,adoptingthesamesimplifiedassumptionas radial flow.
in a few previous studies [7]-[16], we also consider here a
constant mass-shift, homogeneously distributed all over
the system, and related linearly to the asymptotic mass correlation function[7] of K+K− pairs (first and third
by m =m δM. terms in Eq. (1)), as
∗
±
Cs(k1,k2) =1+(E1|+E4E2)2E|c0|2|s0|2 R3e−R2(k12+k2)2+2n∗0R∗3e−(k81m−∗kT2)2 exp −i2mmhuTiR − 8m1T − R2∗2 (k1+k2)2 2
1 2 (cid:12) h(cid:16) ∗ ∗ ∗ ∗ (cid:17) i(cid:12)
(cid:12) k2 k2 −1 (cid:12)
s 2R3+n∗R3(c 2(cid:12)+ s 2)exp 1 s 2R3+n∗R3(c 2+ s 2)exp 2 . (5(cid:12))
× | 0| 0 ∗ | 0| | 0)| −2m T | 0| 0 ∗ | 0| | 0)| −2m T
∗ ∗ ∗ ∗
nh (cid:16) (cid:17)ih (cid:16) (cid:17)io
Themedium-modifiedradiusandtemperatureinEq. (5) shifted massparameter,m , andofthe absolutevalue of
∗
are written, respectively, as R = R T/T and T = their momenta. Therefore, we investigate the behavior
∗ ∗ ∗
T +Asmd2mohnu∗ie2i,natshientcraosdeuocfeφdφincoRrreef.la[t7i]o.nps[15],itisinstruc- otafiCnesd(kb,y−kim,mpo∗s)inagsathfeunidcetaiolinzeodfmlim∗iatnodf|kk1|.=Thiks2is=obk-
−
in Eq. (5). Consequently, a few simplifications occur at
tive to analyze the behavior of the correlation function
for exactly back-to-back particle-antiparticle pairs, i.e., onceinthatequation,i.e.,k1 k2 =2kandk1+k2 0.
− ≡
pairswithexactlyoppositemomenta,asafunctionofthe
3
FIG. 2: (Color online) C(k,−k)×m∗×|k| comparing the instantaneous and the Lorentzian distributions for both the static
case, hui=0, and for an expandingsystem with radial flow parameter hui=0.5.
Another essential assumption is underlying the above Eq. (5). As discussed in Ref. [3], in the adiabatic limit,
result. The solution in Eq. (5) follows when an instan- ∆t , the time factor in Eq. (7) completely sup-
→ ∞
taneous process is considered for the particles’ emission. presses the back-to-backcorrelation(BBC). On the con-
We adopt throughout the paper h¯ = c = 1. In the case trary, in the instantaneous approximation, either from
of instantaneous emission, the time factor is given by Eq. (6) or in the limit ∆t 0 of Eq. (7), the result
→
returns to the form written in Eq. (5), fully preserving
e−i(E1+E2)τ0 2 =1, (6) the strength of the BBC.
| |
The third type of particle emission that we consider
which results from the Fourier transform of an emission
here is a symmetric, α-stable L´evy distribution, i.e.,
distribution described by a delta function. Nevertheless,
it is not expected that it this corresponds to a realistic
F(∆t)2 =exp [∆t(ω +ω )]α . (8)
situation. An emission lasting for a finite time inter- | | {− 1 2 }
val seems more appropriate. Naturally, a priori it is not
This functional form was used in the analyses made by
known which functional form better describes the parti-
the PHENIX Collaboration[18] to fit two- and three-
cle emission process. In what follows, we consider two
particle Bose-Einstein correlation functions. According
other types of distribution. One of them is a Lorentzian
to that analysis, depending on the region investigatedof
form,
the particles’ transverse momentum or transverse mass,
F(∆t)2 =[1+(ω +ω )2∆t2]−1, (7) goodconfidencelevelwasobtainedinthe fit fordifferent
1 2
| | values of α. They found α 1 for 0.2 < m < 0.3 GeV
T
∼
where ω = k2+m2. The Lorentzian emission distri- or α = 1.35 for 0.2 < p < 2.0 GeV/c. Therefore, we
i i T
bution in Eq. (7) was suggested in Ref.[3] and adopted investigate here the time emission factor of Eq. (8) for
p
in previous studies [4, 7, 8],[11]-[16]. Either of the fac- these two values of the distribution index, α. The L´evy
torsin Eq. (6) or(7)shouldmultiply the secondtermin distribution in Eq. (8), should also multiply the second
4
term in Eq. (5). We will see in what follows that the basically wash the effect out. For illustrating the proce-
reduction effect of this distribution on the squeezed cor- duretosearchforthe BBC’s,andsupposingthatNature
relation function is even more dramatic than the effect iskindenoughtoletusenvisagethesqueezingeffectalso
of the Lorentzian in (7). for hadron-antihadron pairs, we restrict our discussion,
We show in Fig. 2 results comparing the time emis- from now on, to the Lorentzian type of distribution.
sion distributions of Eq. (6) and (7). The freeze-out The properties shown in Figs. 2 and Figs. 3 were im-
temperature (T = 177 MeV) and radial flow ( u 0.5) portant for understanding the expected behavior of the
h i≈
parameters were suggested by experimental fits of kaon squeezed correlation function for different values of the
data obtained by the PHENIX experiment [19]. shiftedmass,m ,andback-to-backmomentaofthepair,
∗
Comparing parts (a) and (b) in the top panel of Fig. k1 = k2 =k. This approach,however,focus the study
−
2 with parts (c) and (d) in the bottom, we see that the onthe behaviorofthe maximumvalue ofC (k, k,m ).
s ∗
−
strength of the BBC, C(k, k) m k, decreases al- In other words, if we make an analogy to the HBT ef-
∗
− × ×| |
most three orders of magnitude due the Lorentzian time fect between identical particles, this corresponds to in-
factor,ascomparedtotheinstantaneousemission. How- vestigatethe behaviorofthe correlationfunction’s inter-
ever, the resulting signal is still strong enough to allow cept. Nevertheless, it is not efficient for the purpose of
foritsexperimentalsearch. Anotherinterestingoutcome searching for the BBC experimentally, since the modi-
of the calculation is shown in the left panel in Fig. 2, fied mass of particles is not an observable quantity, ex-
i.e., parts (a) and (c), as compared to the right panel, isting only inside the hot and dense medium. Besides,
i.e., parts (b) and (d). In this case, we see the effect the measurement of particle-antiparticle pairs with ex-
of the expansion of the system on the squeezed correla- actlyback-to-backmomentahaszeroprobabilitytohap-
tion function. The growth of the squeezed correlation pen in practice. It would be more realistic to look for
for increasing values of k is faster in the static case as distinct values of the momenta of the particles, k and
1
| |
compared to when < u >= 0.5, specially at high values k , and combine in an appropriate manner. Therefore,
2
of k. Nevertheless, the presence of flow seems to en- following previous knowledge of identical particle corre-
| |
hance the intensity of Cs(k, k,m ) in the whole region lations (HBT), the first natural tentative method would
∗
−
of the (m , k)-plane investigated, mainly in the lower be tomeasurethe squeezedcorrelationfunction interms
∗
| |
k-region. Naturally, at m∗ = mK± 494 MeV, the of the momenta of the particles combined as their av-
|sq|ueezing disappears, i.e., Cs(k,−k,m∗∼=mK±)≡1. erage, K12 = 12(k1 + k2), and their relative momenta,
In the case of the L´evy distribution, the essential fea- q12 = (k1 k2) [12]-[15]. However, this proposition
−
tures of finite emission interval as compared to the sud- considers non-relativistic momenta and therefore has its
den freeze-out are similar to the ones discussed with re- application constrained to this limit. For a relativistic
gard to Fig. 2. The same is valid when comparing ex- treatment, M. Nagy [12] proposed to construct a mo-
pandingsytemswiththestaticcase,forwhich<u>=0. mentumvariabledefinedasQµback =(ω1−ω2,k1+k2)=
Therefore, we show only results for u = 0.5 in Fig. 3. (q0,2K). In fact, it is preferable to redefine this variable
WcoensciodmerpinagretCha(tk,t−hek)d×urmat∗io×n|kof|ftohrehαem=i is1siaonndcαou=ld1l.a3s5t, raeslaQti2bvbicst=ic l−im(QitµbaisckQ)22bbc=→4((ω21Kω122)−2,KreµtKurµn)i,ngwthoostehenoanv--
either ∆t = 1 fm/c or ∆t = 2 fm/c. We see that, even eragemomentumvariableproposedabove. Althoughnot
for a short-lived system, with ∆t = 1 fm/c and α = 1, invariant,theadvantageofconstructingQ2 asindicated
bbc
the reduction of the squeezed correlation intensity due is that the squeezed correlation function would have its
totheL´evydistributionisevenmoredramaticthanthat maximumaroundthezeroofthisvariable,keepingaclose
due to the Lorentzian time emission. For α=1.35, that analogytotheHBTproceduresandtoitsnon-relativistic
strength is driven to values probably unmeasurable in a counterpart. In the remainder of this paper, we attain
first tentative search. For ∆t = 2 fm/c, the situation is our study to the non-relativistic limit, where the analyt-
considerably worse, even if α = 1. Finally, combining icalresults of the model under discussion, written in Eq.
∆t = 2 fm/c with α = 1.35 reduced the signal basi- (5) and related ones, are safely applicable.
cally to unity, the first non-zero decimal digit being too The analogy with the HBT method is not completely
smallfortheprecisionoftheaxisscale,ifwetriedtoplot transferred to the study of the BBC effect. In HBT ex-
C(k, k) as in the other parts of Fig. 3. That is why perimental analyses a common practice is to replace the
−
in Fig. 3(d) we plot C(k, k) 1. We see that the re- product of the two spectra by mixed events, since these
− −
sulting squeezed correlation function acquires values too are the reference sample not containing statistically cor-
small to be measured by this method. Therefore, if Na- related pairs. However, we see that the second line in
turefavorstheL´evydistributionandiftheemissionlasts Eq. (5), representing the product of the particle and the
a short period, i.e., ∆t=1 fm/c, the predicted strength antiparticle spectra in BBC, does contain the squeezing
ofC(k, k) m k fromFig. 3makesitstillpossible factor f as well. Therefore, the mixed events tech-
∗ i,j
− × ×| |
to search for the signal, if α = 1. However, if α = 1.35, nique would not be an appropriate reference sample in
even if the emission lasts for this short period, it would constructing the BBC correlationfunction.
5
FIG.3: (Coloronline)C(k,−k)×m∗×|k|forthesymmetric,α-stableL´evydistributionwithparametersα=1.0andα=1.35.
Once defined the choice of plotting variables as K Inalltheinvestigationandresultsdiscussedabove,the
12
and q , we can proceed to study the squeezed correla- mass-shifting was considered homogeneously distributed
12
tion function. For emphasizing the characteristics to be over the entire squeezing region, whose size was fixed to
searched for, we focus the study to values of the shifted R=7 fm, the radius of the cross-sectionalarea depicted
mass corresponding to the two maxima located more in Fig. 1. The squeezed correlation function is actually
or less symmetrically below and above the kaon asymp- sensitivetothatsize. Infact,thisisreflectedinitsinverse
totic mass, m = mK± 494 MeV. They correspond to width of the squeezed correlation functions plotted in
∼
m = 350 MeV and m = 650 MeV, respectively. We terms of the average momentum, 2K . In Ref. [15] we
∗ ∗ 12
then calculate the squeezed correlation for K+K− pairs illustratethissensitivitybyconsideringtwovaluesforthe
usingEq. (5). Fromit,iseasilyenvisagedthatweshould radii, R = 7 fm and R = 3 fm. The resulting squeezed
replacek1+k2 =2K12andk1 k2 =q12 inthenumera- correlation function is shown to be broader for smaller
−
tor,atthesametimeasreplacingk =K +q /2,and systems than for larger ones.
1 12 12
k =K q /2 in the denominator. The result ofthis
2 12 12
−
calculation is shown in Fig. 4 and Fig. 5. In both cases
we can observe similar behavior of the squeezed correla- RESULTS FOR K±K± PAIRS
tion functions. The difference resides mainly in the low
q region, where C (K ,q ,m ) reaches much higher
12 s 12 12 ∗ Next, we discuss our findings about the effects of in-
intensities for m = 650 MeV than for m = 350, in-
∗ ∗ medium mass-shift and resulting squeezing on the HBT
cluding for its intercept at K = 0. In both cases we
12 correlationfunction of K±K± pairs. Usual expectations
see that the presence of flow enhances the strength of
were that thermalization would wash out any trace of
C (K ,q ,m ), potentially facilitating its detection in
s 12 12 ∗ mass-shift in these type of correlations. However, as it
an experimental search of the effect.
was demonstrated analytically in Ref. [3, 4], the HBT
6
FIG.4: (Coloronline) Behaviorofthesqueezedcorrelation functioninthe(K12,q12)-plane,fixingthemodifiedmasstom∗ =
350 MeV.
correlation function also depends on the squeezing pa- (with ∆t = 2 fm/c). Finite emission intervals are also
rameter, f (m,m ). described by a Lorentzian distribution similar to that in
i,j ∗
In fact, this identical particle correlation is obtained Eq. (7), obtained as the Fourier transform of an expo-
by inputting in Eq. (1) the chaotic amplitude,
nential distribution in time, but in this case,obtained in
Gc(k1,k2)= E21(2+π)E232 |s0|2R3e−12R2q212+n∗0R∗3(|c0|2+|s0|2) terms of the relative energy, q0 =ω1−ω2, i.e.,
n F(∆t)2 =[1+(ω ω )2∆t2]−1, (10)
×exp − K212 exp − R∗2 + 1 q2 | | 1− 2
2m∗T∗ 2 8m∗T 12 whereω = k2+m2. ThefactorinEq. (10)multiplies
(cid:16) imhuiR(cid:17) h (cid:0) (cid:17) i i i
×exp − K .q , (9) the second term in Eq. (1).
m∗T∗ 12 12 InFig. 6(pa),with∆t=0,nosensitivitytothetwoval-
h io
as well as the expression for the spectrum of each ues of K12 is seen, only the effect of flow is evident. In
particle, G (k ,k ) = Ei s 2R3 + n∗R3(c 2 + the abs|ence|of mass-shift and squeezing, the flow broad-
c i i (2π)32 | 0| 0 ∗ | 0| ens the curves, as expected, since it is well-known that
|s0|2)exp −2mk∗i2T∗ . Since int involves the identical the expansionreduces the size ofthe regionaccessibleto
kaons in(cid:16)this case,(cid:17)othe third term in Eq. (1) gives no interferometry. In part (b), we see that a finite dura-
contribution. Theplotscorrespondingtosuchresultsare tion of the emission separates the curves for each value
showninFig. 6,fortwovaluesoftheaveragemomentum, of K12 , both in presence and in absence of flow. This
| |
K12 =0.5GeV/cand K12 =2.0GeV/c. Theplotsin effectis alsowell-known,andcomesfromthe couplingof
| | | |
the top panel simply illustrate the behavior of the iden- the averagemomentum of the pair to the emissiondura-
tical particle correlation function if no in-medium mass tion, ∆t. Therefore, when there is no mass-shift and no
modification occurs. In (a), for the sudden emission hy- squeezing, the relations describe correctly the expansion
pothesis, and in (b), for emission with finite duration effects on the identical particle correlation function.
7
R = 7 fm T = 177 MeV m* = 650 MeV t = 0 fm/c <u> = 0.5
350
300
q) (b)
K, 250
2
C(s 200
150
100
50 2000
1500
00 1000 - k2
20 k1
40 500 =
2K12= k1+ 6k0 80 100 0 q12
2
FIG.5: (Coloronline) Behaviorofthesqueezedcorrelation functioninthe(K12,q12)-plane,fixingthemodifiedmasstom∗ =
650 MeV.
Whensqueezingispresent,theflowbroadeningisseen not expected to be relevant in this context.
in Fig. 6(c) for K12 = 0.5 GeV/c, but apparently dis-
| |
appears for K12 =2.0 GeV/c. Therefore,it seems that
thesqueezin|geffe|ctstendtoopposethoseofflow,practi- SUMMARY AND CONCLUSIONS
callycancelingthebroadeningofthecorrelationfunction
due to flow for large K . Part (d) essentially repeats In this work we discuss an effective way to search for
12
whatisseenin(c),exc|eptf|ordevisingamodesteffectre- K+K− squeezedcorrelationsinheavyioncollisions,cur-
latedtothefinitedurationoftheemission,whichslightly rently at RHIC, and soon at the LHC. We use suit-
separates the curves corresponding to the two values of able variables introduced previously [8],[10]-[15] to in-
K12 , when u =0. vestigate the expected behavior of the squeezed corre-
| | h i
lation function in an experimental search of the effect.
We remark that we did not include the Coulomb final This is studied by plotting C (K ,q ,m ) in terms of
s 12 12 ∗
state interactions in the above analysis. In the case of the average momentum of the pair, 2K , and its rel-
12
| |
K+K−pairs,eventheGamowfactorwhichover-predicts ative momentum, q . These variables are combina-
12
| |
the strength of the effect for finite distances would be tions of the momenta of the particle and the antiparticle
verysmall. Ingeneral,theeffectoftheCoulombinterac- of each pair, and 2K , is the non-relativistic limit of
12
| |
tions is more pronouncedfor smallvalues of q , which Q2 =4(ω ω KµK ), as discussed previously[12].
| 12| bbc 1 2− µ
corresponds to the region where the hadron-antihadron We started by investigating the general behavior of
squeezing correlation is less favored, therefore being less C (k, k,m ) for exactly back-to-back K+K− pairs, as
s ∗
−
significant to this analysis. Also in the case of K±K± a function of both k and the in-medium shifted mass,
| |
pairs, the squeezing affects the width of the HBT corre- m . This was showing in Fig. 2 comparing the cases
∗
lation function and, since the Coulomb effect is mostly ofsudden particle emission anda finite emissioninterval
concentrated in the region where q is small [17], it is describedby a Lorentziandistribution. A L´evydistribu-
12
| |
8
2.0 2.0
K12 = 0.5 GeV/c K12= 0.5 GeV/c
1.8 K12= 2.0 GeV/c 1.8 K12= 2.0 GeV/c
(), kCk12 1.6 <u> = 0S.5queezinRgt= = = O0 70F Fffmm/c ()Ck, k12 1.6 SqueezinR gt = = = O 0 72F fFfmm/c
1.4 1.4
<u> = 0.5
<u> = 0
1.2 1.2
<u> = 0
(a) (b)
1.0 1.0
0 20 40 60 80 100 0 20 40 60 80 100
q12 (GeV/c) q12(GeV/c)
2.0 2.0
m* = 350 MeV K12 = 0.5 GeV/c m* = 350 MeV K12 = 0.5 GeV/c
1.8 Rt == 70 ffmm/c K12 = 2.0 GeV/c 1.8 t = 2 fm/c K12 = 2.0 GeV/c
= 0 Squeezing ON Squeezing ON
(),kCk1211..46 K<Ku1122> === 200..05 GGeeVV//cc ()Ck,k1211..46 <u> = 0.5 R == 07 fm
1.2 1.2 <u> = 0
<u> = 0.5 (c) (d)
1.0 1.0
0 20 40 60 80 100 0 20 40 60 80 100
q12(GeV/c) q12(GeV/c)
FIG.6: Identicalparticlecorrelation functionsfortwovaluesof|K |,bothforsuddenemission (∆t=0) andforaLorentzian
12
distribution in time, from Eq. (10), with ∆t = 2 fm/c. Parts (a) and (b) show results in the absence of in-medium mass
modification. Parts (c) and (d) consider a shifted mass of m∗ = 350 MeV.
tion was also studied, with results shown in Fig. 3. We enhancetheprobabilityofobservingthesqueezingeffect.
could see the striking reduction effect of finite emission
Another important point discovered within this sim-
intervals,evenforthe Lorentziandistribution. The L´evy
plified model and in the non-relativistic limit considered
type causes an even more dramatic suppression of the
here is that the squeezing could distort the HBT corre-
effect. If this distribution is the one favored by Nature,
lation function as well. It tends to oppose to the effects
the hadronic squeezed correlation function could still be
of flow on those curves,practically neutralizing them for
searched for, if the duration of the emission process is
large values of K .
short, not longer than ∆t 1 fm/c. For longer emission | 12|
≃
time intervals, such suppression would probably destroy Finally,itisworthemphasizingthatthe resultsshown
the effect. here correspond to the signals of the squeezing expected
if the particles have their mass shifted in the hot and
Forillustratingthe proceduretobe followedinthe ex- dense medium formed in high energy collisions. If the
perimental search of the hadronic squeezing effect, we particle’sproperties,suchasitsmass,arenotmodifiedin
suppose that the emission could be considered either themedium,thesqueezedcorrelationfunctionswouldbe
sudden or following a Lorentzian distribution. We then unityforallvaluesof2K,andtherefore,nosignalwould
| |
analyze the behavior of the particle-antiparticle correla- beobserved. Itthatisthecase,thentheHBTcorrelation
tion function, C (K ,q ,m ), in the (K ,q )-plane. functions wouldbehave as usual,both in the presenceor
s 12 12 ∗ 12 12
We find that, in the presence of flow, the signal is ex- absence of flow. However, if the particles’ masses are
pectedto be strongeroverthe momentumregionsshown indeed shifted in-medium, the experimental discovery of
in the plots, i.e., roughly for 0 2K 60 150 squeezedparticle-antiparticlecorrelation(andthedistor-
≤ | | ≤ −
MeV/c (depending on the size of the squeezing region) tions pointed out in the HBT correlations) would be an
and 500 q 2000 MeV/c, suggesting that flow may unequivocal signature of these modifications, by means
≤ | | ≤
9
of hadronic probes. The values of the modified mass, [3] M.Asakawa,T.Cs¨orgo˝andM.GyulassyPhys.Rev.Lett.
m ,adoptedhereforillustratingthesqueezingeffectsfor 83, 4013 (1999).
∗
K+K−pairs,correspondapproximatelytothemaximum [4] P. K.Panda, T. Cs¨orgo˝, Y. Hama, G.Krein and Sandra
S. Padula, Phys.Lett. B512, 49 (2001).
values shown in Fig. 2. However, if the modified mass
[5] M. Gyulassy, S.K. Kaufmann,and L. W. Wilson, Phys.
turns to be shifted away from the maximum values con-
Rev. C 20, 2267 (1979).
sidered in the above calculations, C (2K ,q ) would
s 12 12 [6] A.MakhlinandYuSinyukov,Sov.J.Nucl.Phys.46,354
attain smaller intensities than the ones shown, but the (1987); YuSinyukov,Nucl. Phys.A566, 589c (1994).
signal could still be high enough to be observed experi- [7] Sandra S. Padula, Y. Hama, G. Krein, P. K. Panda and
mentally. The squeezedcorrelationsareverysensitive to T. Cs¨orgo˝, Phys.Rev.C73, 044906 (2006).
the form of the emission distribution in time, as shown [8] Sandra S. Padula, Y. Hama, G. Krein, P. K. Panda and
T.Cs¨orgo˝,Proc.QuarkMatter2005,Nucl.Phys.A774,
above. Instant emissions would fully preservethe signal.
615 (2006).
Lorentziantime distributions woulddrastically reduce it
[9] T.Cs¨orgo˝,B.L¨orstad,andJ.Zima´nyi,Phys.Lett.B338,
andL´evy-typedistributionswouldattenuateitmoredra-
134(1994);P.Csizmadia,T.Cs¨orgo˝andB.Luka´cs,ibid.
maticallyorevenmakethesearchedsignalunmeasurable. B443, 21 (1998).
Anotherimportantpointthatneedsemphasisisthatthe [10] Sandra S. Padula, Y. Hama, G. Krein, P. K. Panda and
squeezed correlation function should be plotted in the T. Cso¨rgo˝, Proc. WorkshoponParticle Correlations and
(K ,q )-plane. If plotted as function of K only, this Femtoscopy (WPCF), AIPConf. Proc. 828, 645 (2006).
12 12 12
means that all the variations in each bin of q are av- [11] T. Cs¨orgo˝ and Sandra S. Padula, Proc. WPCF 2006,
12
Braz. J. Phys. 37 (2007) 949.
eraged out, as they are projected in the K -axis. This
12
[12] Sandra S. Padula, O. Socolowski Jr., T. Cs¨orgo˝ and M.
could enlarge the errorbars and decrease the signal sub-
Nargy,Proc.QuarkMatter2008,J.Phys.G:Nucl.Part.
stantially,dependingontheregionofq selectedforthe
12 Phys. 35, 104141 (2008).
plot. Therefore,theexperimentalsearchforthesqueezed [13] Sandra S. Padula, DanuceM. Dudek and O. Socolowski
hadroniccorrelationsshouldaimatgoodstatisticsofthe Jr., Proc. WPCF 2008, A. Phys. Pol. 40, N. 4, 1225
events for enhancing the chances of its discovery. (2009).
[14] Sandra S. Padula, O. Socolowski Jr. and Danuce
M. Dudek, in Proc. of the XXXVIII International
Symposium on Multiparticle Dynamics (ISMD 2008),
Acknowledgments
DESYPROC200901, 271 (2009), [arXiv:0812.1784v1
(nucl-th)]and[arXiv:0902.0377 (hep-ph),p.271(2009)].
WearegratefultoTama´sCs¨orgo˝andMart´onNagyfor [15] Sandra S. Padula, O. Socolowski Jr., “Searching for
motivating us to investigate the K+K− squeezed corre- squeezedparticle-antiparticlecorrelations inhighenergy
lationsinthe caseofa L´evydistributionofthe particle’s heavy ion collisions”, arXiv:1001.0126 [nucl-th] .
emission as well. DMD is also thankful to CAPES and [16] Danuce M. Dudek, Squeezed Hadronic Correlations of
K+K− pairsinRelativisticHeavyIoncollisions, Master
FAPESP for their financial support during the develop-
Dissertation presented to theInstituto deF´ısica Te´orica
ment of this work.
- UNESP(March/2009).
[17] Miklos Gyulassy andSandraS.Padula,Phys.Rev.C41,
R21 (1990).
[18] M. Csana´d [PHENIX Collaboration], Proc. Quark Mat-
ter 2005, Nucl.Phys. A774, 611 (2006).
∗ Electronicaddress: [email protected], [email protected][19] S. S. Adler et al., PHENIX Collaboration, Phys. Rev.
[1] I.V. Andreev,M. Plu¨mer and R.M Weiner, Phys. Rev. C69, 034909 (2004).
Lett.67 (1991) 3475.
[2] Yu.Sinyukov,Nucl. Phys.A566, 589c (1994).
