Azimuthal angle dependence of HBT radiiwith respect to the Event Planein Au+Au collisions at PHENIX	

TTaakkaaffuummii  NNiiiiddaa  ffoorr  tthhee  PPHHEENNIIXX  CCoollllaabboorraattiioonn  
UUnniivveerrssiittyy  ooff  TTssuukkuubbaa  
WPCF2013 @Acireale, Italy	Space-Time evolution in HI collisions	arXiv:1201.4264 [nucl-ex]	

n Space-time extent at freeze-out reflects the characteristics ofsystem evolution, such as the strength of the expansion, theexpansion time, hadron rescattering, and so on.
n HBT interferometry is a powerful tool to study the space-time
evolution in Heavy Ion collisions. 	

2	

HBT Interferometry	n 1956s, R. Hanbury Brown and R. Twiss measured the angular
diameter of Sirius.

_	n 1960, Goldhaber et al. correlation among identical pions in p+pn Quantum interference between two identical particles
wave function for
2 bosons(fermions) :	

1p=[ (x1 , p1 ) (x2 , p2 ) ±122(x2 , p1 ) (x1 , p2 )]P (p1 , p2 )C2 =⇡ 1 + |˜⇢(q)|2 = 1 + exp( R2 q 2 )P (p1 )P (p2 )R(HBT radius)

spatial distribution ρ	

p1	
detector〜1/R	

r2⇢(r) ⇠ exp()
22R

detectorp2	

q=p1-p2 [GeV/c]	

3	

Azimuthal angle dependence	n Angle dependence of HBT radii relative to Reaction Plane
reflects the source shape at kinetic freeze-out.
n Initial spatial anisotropy causes momentum anisotropy (flow anisotropy)²One may expect in-plane extended source at freeze-outn Final source eccentricity will depend on initial eccentricity,
flow profile, expansion time, and viscosity etc.

Rout-of-plane	
out-of-plane	

aafftteerr  eexxppaannssiioonn  

in-plane	

in-plane	

Rin-plane	

beam	

4	

to the HBT radii from an azimuthally integrated analysis.
Strong 2nd-order oscillations are observed for R2o , R2s ,and R2os , and the signs of the oscillations are qualitatively
self-consistent [10,24], though the amplitude for mostcentral events is small. Similar oscillations were observed

ing Fig. 3 are corrected for momentum resolution following our prescription in Ref. [3].

HBT radii w.r.t Reaction Plane at RHIC	
0.15 < k T < 0.25
0.25 < k T < 0.35central	 STAR PRL93, 012301	
40-80%

2200.40π/2π0Φ (radians)

π/2π+--15

STAR (PRL,93,12301)
εinitial = εfinal

15

PHENIX Preliminary1030

2εfinal2R2l (fm )2R2l (fm )-2+Au+Au 200GeV K K +K K

0.1

020

20.15<kT<0.6 GeV/c	

220

R2os (fm )020

R2os (fm )230+ +

100.2

peripheral	

20Au+Au 200GeV π π +π- π-0.3

1010T. Niida, WPCF2011	

25

2215

200.5

Rs2 (fm )30

Rs2 (fm )Ro2 (fm )252

10-20%

Ro2 (fm )0-5%0.35 < k T < 0.45
0.45 < k T < 0.60

00 10 0.05

0-2
0.1

π/2

0.15

π

0.2

0.25

0π/2

Φ (radians)

0.3π

0.35

0.4

εinitialFIG. 2 (color online). Squared HBT radii relative to the044907 (2004)	
FIG. 1PRC70,(color online).
Squared HBT radii using Eq. (1) relareaction plane angle for four kT (GeV=c) bins, 20%–30%nεfinal ≈εinital/2tive to the reaction plane angle for three centrality classes. Thecentrality
events.
The solid lines show allowed [24] fits to the2 show allowed
2solid lines[24] fits to the individual oscillations.
individual oscillations.

Rs,n = Rs,n (Δφ )cos(nΔφ )2Rs,2ε final = 2 2
Rs,0012301-4

n Strong expansion to in-plane,in-plane	
but still elliptical shape. 	 012301-4

5	

Higher Harmonic Flow and Event Plane	n Initial density fluctuations cause higher harmonic flow vnn Azimuthal distribution of emitted particles:	smooth picture	

Reaction Plane	dN/ 1+2v2 cos2(d+2v3 cos3(2)3)+2v4 cos4(fluctuating picture	

vn = hcosn(Ψ3	Ψ2	Ψ4	4)n )ivn : strength of higher harmonic flowΨn : higher harmonic event plane
φ : azimuthal angle of emitted particles	6	Centrality dependence of vn and initial εn	Higher harmonic flow vn	
Initial source anisotropy εn	
PHENIX	  PRL.107.252301	

PHYSICAL REVIEW C 8

COLLISION-GEOMETRY FLUCTUATIONS AND . . .

1(a)PHOBOS Glauber MC

ε20.501

Au+Au 200GeV

100200ε3ε2v2

PRC.81.054905	(b)Npartv3

Au+Au 200GeV

ε30.50300

PHOBOS Glauber MC

100200300

NpartFIG. 2. Distribution of (a) eccentricity, ε2 , and (b) triangularity, ε3 , as a function of number of participating nucleons,200 GeV Au + Au collisions.

consistent with thefluctuations in the initial state(pexpected
T=1.1GeV/c)	
geometry with the new definition of eccentricity [46]. In thisarticle, we use this method of quantifying the initial anisotropy
exclusively.
Mathematically, the participant eccentricity is given as!"#22 2 + 4(σ )2σy2 − σ3xyx,(3)ε2 =σy2 + σx2

where ψ3 is the minor axis of participant triang

ψ3 =n Weak centrality dependence of v unlike v

atan2(#r 2 sin(3φpart )$, #r 2 cos(3φpa3

It is important to note that the minor axis
is found to be uncorrelated with the react
and the minor axis of eccentricity in Glaub
calculations. This implies that the avera
calculated with respect to the reaction planezero. The participant triangularity defined in E
is calculated with respect to ψ3 and is always
The distributions of eccentricity and trianguwith the PHOBOS Glauber Monte Carlo imp√
for Au + Au events at sNN = 200 GeV areThe value of triangularity is observed to flu
event and have an average magnitude of theccentricity. Transverse distribution of nucleon Initial ε3 has finite values
and weaker centrality
where σ , σ , and σ , are the event-by-event (co-)variances
the participant nucleon distributions along the transverse
dependence than ε2	 ofdirections
x and y [8]. If the coordinate system is shifted to the2x"

2yxycenter of mass of the participating nucleons such that #x$ =#y$ = 0, it can be shown that the definition of eccentricity isequivalent to$#r 2 cos(2φpart )$2 + #r 2 sin(2φpart )$2
ε2 =(4)#r 2 $
3

Triangular component in source shape exists at final state?
➡Measurement of HBT radii relative to Ψ 	
in this shifted frame, where r and φpart are the polar coordinate

7	PHENIX Experiment	
Central
Arm&Magnet

Beam line	

RXNPRXNP	8	Event Plane Determination	
1<|η|<2.8	

24 scintillator segments	

beam pipe	φi	 Ψ3	Ψ2	Ψ4	<cos(n[ Ψn - Ψr ])>Resolution of Event planes

1Event Plane Resolution	
n=2, North+Southn=2, North or Southn=3, North+Southn=3, North or South

0.80.6n Determined by anisotropic flow itself

1tan 1n =n✓⌃wi cos(n i )⌃wi sin(n i )n Event plane is determined byReaction Plane Detector (RXNP)

0.4

0.2

00

beam axis	

102030

40

50

60
70
Centrality [%]²Resolution: <cos(n(Ψn-Ψreal))>n=2 : ~ 0.75
n=3 : ~ 0.34

◆9	Particle IDentification	
EMC here!	
n EMC-PbSc is used.
² timing resolution ~ 600 psn Time-Of-Flight methodm =p2ct
L◆2Particle Identification by PbSc-EMC
2
104

1

p: momentum L: flight path length
t: time of flight	n Charged π within 2σ1.5

Mass square	

2✓!

103

10.5

K-K+	

102

π-π+	

100-0.5-1²π/K separation up to ~1 GeV/c -3-2

-10

1

2

31Momentum × charge	

10	

3D-Analysis	

Rsn “Out-Side-Long” frame² Bertsch-Pratt parameterization² Longitudinal Center of Mass System (pz1=pz2)

C2 =1 + GRlbeam	Ro

λ : chaoticity Rµ : HBT radii	

G = exp( R2 q2 )=exp( Rx2 qx2Ry2 qy2Rz2 qz2⌧ 2 q02 )=exp( Rs2 qs2Ro⇤2 qo2

Rl2 ql2⌧ 2 q02 )LCMS	

⇤222 2(R__________o + T ⌧ )qo=Ro2	

⇡exp(Rs2 qs2

including cross term	

G =exp( Rs2 qs2Ro2 qo2

Rl2 ql2 )Rl2 ql2

22Ros
qs qo )

~kT = 1 (~pT 1 + p~T 2 )2~qo k ~kT , ~qs ? ~kTn Core-Halo model

C2 =C2core + C2halo=N [ (1 + G)Fcoul ] + [1N: normalization factor
Fcoul: Coulomb correction factor	]11	

Correction of Event Plane Resolution	n Smearing effect by finite resolution of the event plane
Reaction Plane	true size	Event Plane	measured size	Reaction Plane	n Correction for q-distribution Acrr (q,Smeared	Corrected!	

Ro2 [fm2 ]Rs2 [fm2 ]ü model-independent correction²Checked by MC-simulation
25j)+2⌃⇣n,m [Ac cos(n j ) + As sin(n j )]n /2⇣n,m =sin(n /2)hcos(n( mreal ))i²PRC.66, 044903(2002)30j ) =Auncrr (q,event plane resolution	

30

w.r.t RPw.r.t RP

25202015151010

Uncorrected w.r.t EP

00.5

1

1.5

2

2.5

3

Δφ [rad]Uncorrected w.r.t EPCorrected w.r.t EP
00.5

1

1.5

2

2.5

3

Δφ [rad]12	

HBT radii w.r.t 3rd-order event plane	
R2o [fm2]

R2s [fm2]

Average of radii is set to “10” or “5” for w.r.t Ψ2 and w.r.t Ψ3	10 105
Au+Au 200GeV 0-10%π+π++π-πw.r.t Ψ250.2<kT<2.0 GeV/c	0-3-2-10123

φ - Ψn [rad]T. Niida (QM2013)	

w.r.t Ψ30-3-2-10123

φ - Ψn [rad]n Ro clearly shows a finite oscillation w.r.t Ψ3 in most central event,while Rs does not show such a oscillation.
□ Δτ depends on azimuthal angle?
n What makes this Ro oscillation?
Note: Ro is sensitive to Δτ & βT"
C2 =1 + exp( Rs2 qs2 Ro2 qo2 Rl2 ql2
Ro2 =Ro⇤2 +2
T22Rosqo qs )⌧2

□ effect of flow anisotropy?
□ difference of “width” and “thickness”?

13	

oscillation amplitude for Ro is much larger than for Rs .
As already discussed at the end of Sec. II, the observed
2small triangular oscillation amplitude Rs,3of Rs2 cannot[14] be directly interpreted as evidence for a small geometric triangularity of the source at freeze-out. What,then, is the correct interpretation of the experimental
observations?
PRC88, 044914 (2013)	 nPossible explanation	

25

R2o

R2s, o "fm 2 #20

Thick blue: deformed flow field
deformedflow
Thin red : deformed geometry
deformed geometry	R2s15105%Π

HBT radii w.r.t Ψ3 with Gaussian model²C. Plumberg et al., PRC88, 044914 (2013)²Next talk: C. Plumbergn with/without triangularly deformed
flow/geometry	
deformed flow	

flow	

K! !0.5 GeV%2Π!3

deformed geometry	%Π!30

Π!32Π!3

Π'%$3

FIG. 3: (Color online) Triangular oscillations of Rs2 (dashed)geometry	and Ro2 (solid) for pion pairs with momentum K⊥ = 0.5 GeV,
as a function of emission angle Φ relative to the triangularflow direction Ψ3 . Shown are results for two model scenar“Deformedfinite
Ro oscillation and very small Rs oscillation
ios: A n
deformedflow field (v̄3flow”= 0.25) showsin a spatiallyisotropic(¯"3 = 0) density distribution (thick blue lines), and a sourcewith triangular
geometric deformation(¯"3 = 0.25)with
expandingn Qualitatively
agreement
the data seen in most-central collisions
with radially symmetric (v̄3 = 0) flow (thin red lines). For thetwo scenarios the oscillations of both Rs2 and Ro2 are seen to
be out of phase by π/3.

In Fig. 3 we show triangular oscillations relative to14	

1515151030k Tkt:0.2-0.35[GeV/c]0.2-0.35
kt:0.46-0.58
0.46-0.58
0.75-1.5
kt:0.75-1.5

030

0.5

11.5

22Rµ2 = Rµ,0+ 2Rµ,3cos[3(- -+ +Au+Au 200GeV
π- π- +π π+ +Au+Au 200GeV / / +/ /2

φ - Ψ [rad] 
300.5

11.5

2

φ - Ψ 3 10[rad]R2os=0R2 =0l55

kt:0.35-0.46
0.35-0.46
kt:0.58-0.75
0.58-0.75

os

1022Ros= 2Ros,3sin[3(3 )]3 )]µ = s, o, ln No clear kT dependence for Rs

2020

R2os [fm2]2RR2l 2[fm][fm2]

55n Charged pions in Au+Au 200 GeV²20-60% centrality
10²5 kT bins within 0.2-1.5 GeV/cn Fitted with the following Eq.:

10

10

R2o [fm2]

20-60%

15

R2o [fm2]

20-60%

R2os [fm2]s 2]
R2s [fmR2 [fm2]kT dependence of HBT radii w.r.t Ψ3	5510n Same sign of the Ro oscillation in allkT bins

10
000.5

100.5

11.5

2q - ^ [rad]31.5

20φ - Ψ [rad]
30.5

01

0.5

1.5

211.5q - ^ 3 [rad]02φ - Ψ 3 [rad]15	R2s,3/R2s,0mT dependence of 3rd-order oscillation amplitudes	
0.05

Au+Au 200GeV /+/++/- /0-20%

Rs20-60%

Ro

00.05

R2o,3/R2o,00.05

 

00.05
-0.1

0.5

1

<mT > [GeV/c]

0.51<mT > [GeV/c]n Rs,32 are around zero, and does not show any clear mT dependence.
n Ro,32 has finite negative values in both centrality²In 20-60%, it seems to decrease with mT.

16	R2s,3/R2s,0Comparison with the 3rd-order Gaussian model	
0.05

Au+Au 200GeV /+/++/- /0-20%

Rs20-60%

Ro0

deformed flow	
0.05

flow	
PRC88,044914
deformed flow
deformed geometry

R2o,3/R2o,0

0.05
00.05

deformed geometry	geometry	 

-0.1

0.5

1

<mT > [GeV/c]

0.51<mT > [GeV/c]n Trend of Ro,32 seems to be explained by “deformed flow” in both centralities.
²Note that model curves are scaled by 0.3 for the comparison with the datan Rs,32 seems to show a slight opposite trend to “deformed flow”.
²Zero~negative value at low mT, and goes up to positive value at higher mTn Contribution from spatial anisotropy seems to be small.

17	Eccentrici5es'vs'hiTimeevolution of spatial
anisotropy	
Eccentrici5es'vs'higher'harmonics

MC-KLN + Hydrodynamic model• RHIC)(200)events)
• nRHIC)(one)event)
²Parameters are not tuned.

n 15-20% centrality	

εv

εn

eps1
eps2
eps3
eps4
eps50.8

1 event	

eps1
v1
eps2
v2
0.80.2 eps3
v3
eps4
v4
eps50.60.15 v5200 events	

¡vnn

¡n

0.6

10.25

vn1

C. Nonaka, Aug. 2013,@2nd workshop on
initial fluctuationsnn
and final correlation	

0.4

0.40.1

0.2

0.2
0.05

001"

2345time (fm)678

90 00 01 12 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9time(fm)time(fm)

Inflection points represent that the nth-order deformation of thesource turns over.
" Interesting that ε3 turns over earlier than ε2.
C.)NONAKA)
C.)NONAKA)

18	

Summary	n Azimuthal angle dependence of HBT radii with respect to 3rdorder event plane have been presented.
²Finite oscillation of Ro2 and very weak oscillation of Rs2 seen in
most central event may be explained by the triangular flow
anisotropy rather than spatial anisotropy.
²Ro,32 shows a monotonic decrease with mT.
ü Similar trend to “deformed flow” model²Rs,32 does not show any clear mT dependence, but seems to have
opposite trend to “deformed flow” model.
n The result indicate that initial triangularity may be significantly
diluted.

19	Thank you for your attention!	

20