FFiinnaall  ssoouurrccee  eecccceennttrriicciittyy  mmeeaassuurreedd  bbyy

  HHBBTT  iinntteerrffeerroommeettrryy  wwiitthh  tthhee  eevveenntt  sshhaappee  sseelleeccttiioonn

Takafumi Niida
for the PHENIX Collaboration
University of Tsukuba
Hot Quarks 2014
Las Negras, Spain

Event shape engineering with HBTat the PHENIX experiment

CCoonntteennttss
Event twist selection with HBTwith AMPT model2

EEvveenntt  sshhaappee  eennggiinneeeerriinngg
2,alower 5%

3

10

0.6
102

(a) 0.474
2000
B! Q2,x = ⌃wi cos(2 )Q2,x= ⌃wi cos(2 )! Q2,y= ⌃wi sin(2 )Ψ2Q2,y! = ⌃wqpi sin(2 )2 + Q2 / M
q
Q
=
Q
p
2
2,x
2,y
!
2
2
Q2 = Q2,x + Q2,y / M!
1 Q2,y=tan()
2
1 Q2,y)! 2 = tan ( Q2,xQ2,x!
Possibly control the initial geometry

2,a

q

10
0.4

1500
F

Selecting e-b-e v2 by the magnitude of ﬂow vector

no cuts
q upper 5%

104

∈F2

J. Schukraft et al., arXiv:1208.4563

Centrality 20-25%

1

0.2 0

1000

0.05

0.1

0.15

0.2

0.25

0.3

v2500

initial shape
ε2events∈F2number ofEvent shape engineering (ESE)number of events

J.Schukraft et al., arXiv:1208.4563

0
0

0.2

0.4

0.6

Centrality 20-25%
J.Jia et al., arXiv:1403.6077

∈B2

104
0.6

no cuts
q3,a upper 5%(c) 0.640

q3,a lower 5%

4000

103
3000

!
More accurate connection between initial
and ﬁnal source eccentricity ?
Azimuthally sensitive HBT w.r.t Ψ2

2000

10
0.2
AMPT
Pb+Pb 2.76TeV

1
00
0

0.05

0.1
0.1

0.15

qF

0.2
0.2

1000

0.25 0.3
0.3

ﬂow vector q22v3(e) online)
0.354Actual (true) v2 and v3 distributions
FIG. 3. (color
in the event
samples selected by di↵erent cuts on the corre2
sponding qn -vector magnitude indicated in the plot compared
2000
to that extracted from the BG fits to qn,b distribution shown
B
2* )

3

B

0.42
10

HHaannbbuurryy  BBrroowwnn  aanndd  TTwwiissss  iinntteerrffeerroommeettrryy
"
tor
c
e
det

p1

R"

Hanbury Brown and Twiss eﬀect (1950s)

tor"
c
e
t
e
d

(HBT"radius)"

Quantum interference b/w two identical particles
Due to (a)symmetrization of the wave function ofidentical bosons(fermions)p2

q =q =
p1p1 p2p2!
P (p1 , p2 )C2 ! =⇡ 1 + |˜⇢(q)|2 = 1 + exp( R2 q 2 )P (p1 )P (p2 )!
Experimentally correlation function

HBT$eﬀectNreal (q)C2 =Nmixed (q)!
!

Coulomb$repulsion$$+$contamina6onEnhancement at low q by HBT eﬀect
including ﬁnal state interaction (Coulomb, strong)

Rs#
beam

Nreal: real pairs in the same event
Nmixed: pairs made by event mixing

Bertsch-Pratt parameterization w/ core-halo picture

Rl#
Ro#

C2 =C2core + C2halo=[ (1 + G)Fcoul ] + [1~kT = 1 (~pT 1 + p~T 2 )2~qo k ~kT , ~qs ? ~kT

G =exp( Rs2 qs2
4

Ro2 qo2]
Rl2 ql22
2Ros
qs qo )

PPHHEENNIIXX  eexxppeerriimmeenntt
Centrality, zvertex
Beam Beam Counter (3<|η|<3.9)

EMC
al
PC
DC

Beam line

BBC

R
X
N
P

!
Event plane & ﬂow vector determination

R
X
N BBC
P

Reaction Plane Detectors (RxNP) (1<|η|<2.8)
Res(Ψ2)~75%

Tracking
Drift Chamber + Pad Chambers (|η|<0.35)!
Charged pion identiﬁcation1

n=2, North+Southn=2, North or Southn=3, North+Southn=3, North or South

0.8

Electromagnetic calorimeter (EMCal) (|η|<0.35)
using time-of-ﬂight
at EMCal
Particle Identification by PbSc-EMC
2
104

0.6

Mass(square<cos(n[ Ψn - Ψr ])>Resolution of Event planes

Event Plane Resolution

0.4

0.2

0
0

1.5

103

1

0.5

20

30

40

50

102

π"#########π+

10

0

-0.5

10

K"#########K+

60
70
Centrality [%]
-1
-3

5

-2

-1

0

1

Momentum(×(charge

2

3

1

HHooww  ttoo  aappppllyy  tthhee  EESSEE
1. Q2 distribution measured by RxNP
2. Fitted with the Bessel-Gaussian functionxx0 x(x20 + x2 )fBesselGaus = I0 ( 2 ) exp()
2
2
3. Select higher or lower Q2 events

Resolutions of event planes were
estimated by 3-sub method using
RxNP(1<|η|<2.8) and BBC(3<|η|<3.9)
applying Q2 selection.

6

CChhaarrggeedd  hhaaddrroonn  vv22  wwiitthh  EESSEETest of the event shape engineering for v2 in Au+Au 200GeV collisions
v2 measured at mid-rapidity (|η|<0.35)
Q2 and EP determined at 1<|η|<2.8

Conﬁrmed that higher(lower) Q2 selects larger(smaller) v2

7

HHBBTT  rraaddiiii  ww..rr..tt  Ψ22  wwiitthh  EESSEE  ~kT = 1 (~pT 1 + p~T 2 )2~qo k ~kT , ~qs ? ~kT

Rs#
Rl#

beam

Ro#
r
e
v
r
e
obs

Applying the ESE to azimuthal HBT22
Rµ2 = Rµ,0
+ 2Rµ,2cos(2charged ππ-correlation measured at mid-rapidity (|η|<0.35)
Q2 and EP determined at 1<|η|<2.8

Oscillations of Rs and Ro become larger when selecting higher Q2
except Ro in 0-10%8)εﬁnal ~ 2Rs,22/Rs,02
F. Retiere and M. A. Lisa, PRC70.044907
at the limit of kT=0!

2Rs,22 / Rs,02

FFrreeeezzee--oouutt  eecccceennttrriicciittyy  vvss  NNppaarrtt  wwiitthh  EESSEEHigher Q2 selection increases the

ε2finalmeasured εﬁnal
Selected more elliptical source at freeze-out?
might be originated from εinit
Or just v2 eﬀect?

92Ro,22 / Ro,02

Rside modulation 
Rout modulation 

ε2final

EEvveenntt  ttwwiisstt  sseelleeccttiioonn  wwiitthh  HHBBTT

wwiitthh  AAMMPPTT  mmooddeellTTwwiisstteedd  ssoouurrccee??
3

3shouldexpectfollowing
relation
to hold:
weweshouldexpectthe thefollowing
relation
to hold:

Twisted ﬁreball due the density ﬂuctuationv⃗n (η)≈ c≈ϵFϵBv⃗n (η)cn[α(η)⃗(η) [α(η)⃗ϵF(1 − α(η))⃗ϵBn (η)
n + (1n]n −+α(η))⃗
n]of wounded
nucleons
going(10)to forward
⃗geo
⃗dyn,
+δ⃗geo (η)+ δ⃗dyn (η)
n +δn(η)n + δn (η) ,

(10)

and
directions
where
thethecbackward
is thehydrodynamic
responsefunction,
n (η)
where
cn (η)
is thehydrodynamic
responsefunction,and the three additional terms in the form of δ⃗n = δn⃗einσn inσnand thethree
additionalterms
in the form of δ = δ eP.additional
Bozek
et al.,
PRC83.034911
represent
initialor finalstate eﬀects. The nterm n
represent
additional
initial geometricor final stateeﬀects.
Thetermδ⃗ngeo(η)
represents
additionaleﬀects
not
acgeo
J.
Jia
et al.,additional
arXiv:1403.6077
δ⃗n (η)eﬀectscountedforrepresents
by the eccentricity,
suchgeometricas the details
in thenot accounted
for by the
eccentricity,such profileas the [15,details
radialdistributionof theenergy density
39] in the
Also
known
“eventplaneradialdistribution
thealternative
energy density
profileandthe diﬀerence
fromofasandefinitionof decorrelation"ec-[15, 39]
and the[20].
diﬀerence
alternative
The lastfrom
termanδ⃗ndynrepresents definition
additional of eccentricity
dyn
K.
Xiao
et
al.,
PRC87.011901⃗
last41]term
δn represents
centricity[20]. The[40,dynamical
fluctuations
generated
during theadditionalhydrodynamic
evolutionand hadronization.
dynamical
fluctuations[40,41] generated
duringthe hydecorrelation
increaseswithincreasing
η-gap
F
B
Secondly,ϵ⃗n andϵ⃗n fluctuatestrongly event to event,drodynamic
evolutionand hadronization.
both Secondly,in their magnitude
and orientation. If ϵF≠ ϵBn event
n , theϵ⃗F⃗B
to event,n and ϵn fluctuate strongly
distributionsof flow
coeﬃcientsto Bn (η) are expectedboth in theirmagnitudeand vorientation.
If ϵFn ≠ ϵn , theshowstrong forward-backward
asymmetry.
if
distributionsof flow
coeﬃcientsvn (η)Similarly,
are expected
to meansv
maybeunderestimated,
which
n
∗F
∗B
Φn ≠ Φn , the event-plane angle Φn is expected to rotate
show strong forward-backward asymmetry. Similarly, if
graduallyfrom∗F∗B backward rapidity to the forward rapidoverestimating
η/s
Φ
≠
Φ
the event-planeangle Φn function,is expected
to rotaten ,since
ity. nHoweverα(η) is a non-linear
theseF
gradually
backward
rapidity
to theforwardrapidchanges
may from
also notbe linear,
especially
when
Npart
ity.
BHowever since α(η) is a non-linear function, these
and
Npart
values are very diﬀerent such as in Cu+Au or
Fchanges
may also notB be linear, especially when Npart
p+Pb collisions.
Ψ2
B
and
N
valuesarediﬀerentin Cu+Au
or
A simple
Glaubermodelsuch[42] asis usedtopartmonte-carlo very
estimate
FB eccentricity fluctuations in Pb+Pb colp+Pb thecollisions.
lisions.
results
as a functionof Npartmodel
are summarized
A The
simplemonte-carlo
GlauberF is used toΨ2[42]
FB
of ϵ2 and ϵ2 arefound to bein estimate
Fig. 2. The
thevalues
FB eccentricity
fluctuations
in always
Pb+Pb collarger
thanTheϵ2 overthe full(Figure2(a)),
lisions.
results
as acentrality
functionrangeof Npart
are summarized
andis dueoftoϵF
the factBthat the center-ofz"di be always
2. The values
in thisFig.diﬀerence
2 and ϵ2 are found toc(omass of the wounded nucleons in each nucleus is notreatnlarger
than
ϵ
over
the
full
centrality
range(Figure2
11 of the overlap region but is shifted towards 2(a)),
the center!

FIG. 1: Schematic illustration of the forward-backward flucarXiv:1403.6077

tuation of second-order eccentricity and participant plane, intransverse
plane (a)and along of
rapidity
B
Fdirection (b) in A+A
FIG.
1: Schematic
illustration
the forward-backward
fluccollisions.
The
dashed-lines
indicatetheparticle
production
tuation of second-ordereccentricity andparticipant plane, in
part
part
profiles forforward-going
and rapidity
backward-goingtransverse
plane(a)B and along
directionparticipants,(b) in A+AFFB
f (η)Npartandf (η)Npart
respectively.
B ,indicate
Fthe particle production
collisions.
The
dashed-lines

N

6= N

" 6= "

nprofiles for forward-goingnand backward-goingparticipants,
FF
B
B
f (η)Npart and f (η)Npart , respectively.
B
F
where 22where α(η)=

part,n 6=

part,nF
f F (η)Npart⟨rn ⟩F,(9)and this diﬀerence is due to the fact that the center-of-

0.6

0.6

0.6
3000

EEvveenntt  ttwwiisstt  sseelleeccttiioonn
2000

0.2

0.4

0.2

qF

0.2

0
0

0.3

0.1

(a)

2

(e) 0.354

0.5

2

S
F
q2 bin = 24,q

0.2

J.Jia et0 al., arXiv:1402.6680

0.3
cut

0

(f)

1.05
3

1

F
B
2(
Ψ
Ψ
)
2
2B
F

4
-2

2

2 0

2(Φ * -Φ * ) 0
F2∆ φ2(Ψ2 -Ψ2 ) final twist

B

2

0.15

∆ η∈ [5,6]∆ η∈ [-4,-3](e) 0.125∆ η∈ [3,4]∆ η∈ [-2,-1]∆ η∈ [1,2]

0

10

-2

1

AMPT Pb+Pb 2.76TeV

1

-5

0

∆η

1.05

-0.5
-2

C(∆ φ)

0C(∆ φ)∈F2-∈B2

-2

5

1000

102

0.1

qF3
∆ η∈ [-6,-5]

2

10

2000

0

0.05

(b)

top 10% of Ψ20

initial twist
B*B)
2(Φ22*F*F
-Φ-Ψ
2(Ψ
2* )2

0.1

0.4

0.2

1000

1000

0
0

∈F3

2000

3(Φ3*F-Φ3*B)

0.4

∈B2

∈F2

3000

-2

0

23(Ψ F3-Ψ B
3)

0

2

2
4

∆φ

a b
rot
C(
,
⌘)
/
1
+
2⌃v
v
cos(n
n
FIG. 8: The 2D correlation function (left),n1Dncorrelation function in di↵erentn )⌘ sF in the fourth q S bin with largest cut valu
nF rotfor n2∆Φ= 2–5∗FB(right),for ϵevents
selectedn(a),FIG. 4: Correlation of ϵB
vs.
ϵ
vs.
(b),
ϵ
FIG. 5:2 Correlation 2of ϵB
2
2
2
2
2
3
F
F
∗FB
FB
B
F
F
−
ϵ
(e),
and
ϵ
vs. q2B
(d),
2∆Φ
vs.
2∆Ψ
vs.
q
(d),
vs. q2F Twist(c), ϵF
vs.
q
(c),
ϵ
2
2
2
2eﬀect3
3
3
2
on
anisotropic
ﬂow&2PC
studied
with
AMPT
∗FB
q bin = 10, top 10% of Ψ(a)(b)∆ η∈ [-6,-5]∆ η∈ [5,6]vs. 2∆Φ∗FB
(f)
for
AMPT
Pb+Pb
events
with
b
=
8
fm.
The
vs.
3∆Φ
(f) for
AMPT
B
F
2 Requiring ﬁnite diﬀerence b/w forward and backward EPs (Ψ -Ψ )322
∆ η∈ [-4,-3]∆ η∈ [3,4]bars around the circles in panel-(b) indicates the RMS widthbars aroundthe circlesin∆ η∈ [-2,-1]∆ η∈ [1,2]1.1∗FB eﬀect appears as a phase shift in Δφ-Δη correlation∗FBof 2∆ΦTwist
at
given
ϵ
value,
andthefour
regionsdelineatedof3∆Φat given ϵ3 va
2
23initialintwist
survives
as a ﬁnal
state
ﬂow
momentumspace by the boxes in panel-(f)by the boxes
panel-(f)indicate
thecuts
forinthefour eventclassesHow
definedin Tableclasses defined in Table I
I. Thenumbers inspace?
some panels indiabout
in spatial
coordinate
1cates the correlation coeﬃcients of 1.1
the distributions.
cates the correlation coeﬃ
12
cut
2

∆η

C(∆ φ)

5

0

1

C(∆ φ)S
2

HHBBTT  ssttuuddyy  iinn  AAMMPPTT  AMPT modelver.2.25 (string melting)
Pb+Pb 2.76 TeV collisions, b=8fm
initial ﬂuctuation based on Glauber model and ﬁnal state interaction via transportmodel

EP determination at 4<|η|<6

!
!
!

HBT!
"3.5<η<3.5!
Δη=1!step←twist!cut!
←a!reference!angle!of!HBT

!"6!!"5!!"4!!"3!!"2!!"1!!!0!!!1!!!2!!!3!!!4!!!5!!!6

η

HBT analysis
Add HBT correlation between two pion pairs(1+cos(ΔrΔq)) was weighted when making q-distribution of real pairs

Allowing to take π+π- pairs to increase statistics
conﬁrmed a good agreement between π+π+ and π-π-

No EP resolution correction
Bertsch-Pratt parameterization

C2 = 1 + exp( Rs2 qs2Ro2 qo2

Rl2 ql2
13

2
2Ros
qo qs

2
2Rol
qo ql

2
2Rsl
qs ql )

9

7

bbaacckkwwaarrdd
15

R 2l [fm2]

AMPT Pb+Pb 2.76TeV
b=8fm

R 2o [fm2]R 2s [fm2]

HHBBTT  rraaddiiii  ww..rr..tt  bbaacckkwwaarrdd  Ψ22
(ΨB2- ΨF2)>0.6

6

Ψ2B

10

Ψ2F

8
5
7
1

2

2.5<η<3.5-2.5<η<-1.5

1.5<η<2.5

4 0

3∆ φ [rad]

1

2

3

on

ffoorrwwaarrdd
0

R 2sl [fm2]Selected events with (Ψ2B-Ψ2F)>0.6

z"di
rec(

5

∆ φ [rad]R 2ol [fm2]R 2os [fm2]0-3.5<η<-2.5

1

2

3

∆ φ [rad]2

AMPT Pb+Pb 2.76TeV π π -HBT
Phase shift can be seen, and data are ﬁtted with cosine(sine)
0.2<k T<1 GeV/c, EP: -6<η<-41
1
1function including a phase shift parameter α

0

-1

2
0 2
Rµ2 = Rµ,0
+ 2Rµ,2 cos(2
-1

14

+ ↵)

0
-1

-3.5<η<-2.5-2.5<η<-1.5-1.5<η<0.5

2.5<η<3.5
1.5<η<2.5
0.5<η<1.5

97bbaacckkwwaarrdd
15

R 2l [fm2]R 2o [fm2]R 2s [fm2]  
HHBBTT  rraaddiiii  ww..rr..tt  ffoorrwwaarrdd  Ψ22

(ΨB2- ΨF2)>0.6

6

Ψ2B

10

Ψ2F

8
5
7
1

2

2.5<η<3.5-2.5<η<-1.5

1.5<η<2.5

4 0

3∆ φ [rad]

1

2

3

on

ffoorrwwaarrdd
0

R 2sl [fm2]Selected events with (Ψ2B-Ψ2F)>0.6

z"di
rec(

5

∆ φ [rad]R 2ol [fm2]R 2os [fm2]0-3.5<η<-2.5

1

2

3

∆ φ [rad2

AMPT Pb+Pb 2.76TeV π π -HBT
Phase shift can be seen, and data are ﬁtted with cosine(sine)
0.2<k T<1 GeV/c, EP: 4<η<61
11function including a phase shift parameter α

022Rµ2 =2Rµ,0+2 2Rµ,2cos(202R =2Rµ,0 + 2Rµ,2 cos(22 µRµ = 2Rµ,2 sin(2
-1

+ ↵)

+ ↵)

-1

15

+ ↵)

0
-1

-3.5<η<-2.5-2.5<η<-1.5-1.5<η<0.5

2.5<η<3.5
1.5<η<2.5
0.5<η<1.5η--ddeeppeennddeennccee  ooff  pphhaassee  sshhiifftt
22
Rµ2 = Rµ,0
+ 2Rµ,2
cos(2

0.4
0.2

+ ↵)

0.6

αos [rad]0.6

αout [rad]αside [rad]22Ros= 2Ros,2
sin(2

+ ↵)

0.4

0.6
AMPT Pb+Pb 2.76TeV
0.4

0.2

0.2

0

0

0

-0.2

-0.2

-0.2

-0.4
-0.6

-0.4

-0.4

Ψ22BB
-3

-2

-1

0

1

2

-0.6

3

w.r.t ΨB2
w.r.t ΨF2

Ψ22FF
-3

-2

-1

0

1

2

η

-0.6

3

-3

-2

-1

0

1

η

2

3

η

Phase shifts become larger with going far from η of a reference EP(-6<η<-4 or 4<η<6)Source at freeze-out might be also twisted as well as EP angles
It may include the eﬀect from twisted ﬂow

This twist eﬀect could be measured experimentally16

SSuummmmaarryy
Event shape engineering at PHENIX
Azimuthal HBT measurement with the event shape engineering have been
performed in Au+Au 200GeV collisions
Higher Q2 selection enhances the measured εﬁnal as well as v2
More accurate relation between initial and ﬁnal eccentricity
Applicable for detailed study like a path-length dependence in 2PC?

!
Event twist selection with AMPT model
A possible twisted source have been studied via HBT measurement with AMPT
Pb+Pb 2.76TeV collisions
Phase shifts of HBT oscillations are seen as a function of η, possibly
indicating the twisted source at ﬁnal stateThis eﬀect might be measured in RHIC and the LHC, especially in ATLAS or
CMS!
These technique might be useful for Cu+Au and U+U
17

TThhaannkk  yyoouu  ffoorr  yyoouurr  aatttteennttiioonn

Rside : width

52

pair and ΨEP, 2

SSppaaccee--mmoommeennttuumm  ccoorrrreellaattiioonn
Rout : depth

2 cos(2('+emission duration

2 ))Emission points of pions in the Blast-wave model (PRC70, 044907)
HBT radii = “length of homogeneity”
known))
as kT dependence of HBT radii by radial flow

out-of-plane

2

PHYSICAL REVIEW C 70, 044907 (2004)

20

mass andcles withs that are

t

14

in

,0

observerin-planepo

Becauseon points
us of thee average
mit. This
described
e particle

βx＝0.974
ew

4:

24

βx＝0.907vi

cos(2('
ands,2 kaonsomentum.
rameters "T = 0 GeV.
= 0.1 GeV.
e: Rx = Ry

2nd order
Event plane
FIG. 37. Distribution of the emission points in the transverse
Azimuthal plane
plane, for different particle species emitted at the same velocity,
19'y = 0 and 'x = 0.907 on the left-hand side and 'x = 0.974 on theHHBBTT  rraaddiiii  ww..rr..tt  Ψ22  wwiitthh  EESSEE((RRll  aanndd  RRooss))

Oscillation of Rl doesnʼt change, while Ros increases when selecting
higher Q2 events as well as Rs and Ro

20

EEvveenntt--bbyy--eevveenntt  vvnn  aatt  AATTLLAASS
2102

pT>0.5 GeV, |h|<2.5

pT>0.5 GeV, |h|<2.5

Pb+Pb

ATLAS Pb+Pb

ATLAS Pb+Pb

sNN=2.76 TeV

sNN=2.76 TeV-1Lint = 7 µb-1

ATLAS

10

sNN=2.76 TeV
Lint = 7 µb

-1

10Lint = 7 µb

pT>0.5 GeV, |h|<2.5

4

p(v )

p(v 3)

2

p(v )

1
10

1
-1

10

10-2
0

centrality:
0-1%
5-10%
20-25%
30-35%
40-45%
60-65%

10-1
0.1

0.2

0

v2

centrality:
0-1%
5-10%
20-25%
30-35%
40-45%

centrality:
0-1%
5-10%
20-25%
30-35%
40-45%

0.05

v3

0.1

1
0

0.01

0.02

0.03

0.04

0.05

v4

dN
∝1+ 2∑ v n cos n φ − ΦFigure
10. The probability density distributions of the EbyE vn in several centrality intervals fondφnn = 2 (left panel), n = 3 (middle panel) and n = 4 (right panel). The error bars are statisticaΦn)?

ear) dynamics?

mics?

(

)

uncertainties, and the shaded bands are uncertainties on the vn -shape. The solid curves are distri
butions calculated from the measured ⟨vn ⟩ according to eq. (1.6). The solid curve is shown only fo
0–1% centrality interval for v2 , but for all centrality intervals in case of v3 and v4 .

from ref. [31]. However, the uncertainties from ref. [31] for the vn -scale (table 2) andvn -shape, are well within the total systematic uncertainties derived from the data analysis
21

OOsscciillllaattiioonn  aammpplliittuuddeess  aass  aa  ffuunnccttiioonn  ooff  NNppaarrtt

wwiitthh  EESSEE22

R 2o [fm2]12

R 2s [fm2]1412

R 2l [fm2]BB
HHBBTT  rraaddiiii  ww..rr..tt  Ψ22
(ΨB2- ΨF2)>0.6

15

10

10

8

8

10

6

6

5

4
2

3

0

∆ φ [rad]

1

0

1

2

3

1

0

-1
1

2

3

∆ φ [rad]1

2

3∆ φ [rad] [GeV/c]
0.2<kT<1
0.2<kT<1 [GeV/c]

2
1

AMPT Pb+Pb 2.76TeV π π -HBT
0.2<k T<1 GeV/c, EP: -6<η<-40-3.5<η<-2.5-2.5<η<-1.5-1.5<η<0.5-0.5<η<0.5

-1

-1
0

0

∆ φ [rad]R 2sl [fm2]1

R 2ol [fm2]R 2os [fm2]

4 0

0

1

2

3

∆ φ [rad]

-2

0

1

2.5<η<3.5
1.5<η<2.5
0.5<η<1.5

2

3∆ φ [rad]Selected events with (Ψ2B-Ψ2F)>0.6
Phase shift can be seen, and become larger with going far from ηof EP for a reference angle (-6<η<-4)23

14

R2long [fm2]R2out [fm2]R2side [fm2]BB((FF))  
HHBBTT  rraaddiiii  ww..rr..tt  Ψ22
((η<<00))
10

12

8

AMPT Pb+Pb 2.76TeV

10

10
6
8
5

4

2

0
0 0.5

1 1.5 22.5

3

∆ φ [rad]1 1.5 2

2.5

3

0 0.5

∆ φ [rad]2

R2sl [fm2]R2os=04

0 0.5

R2ol [fm2]R2os [fm2]

6 0 0.5 1 1.5 2 2.5 3∆ φ [rad]1

1 1.5 2

w.r.t Ψ B (-6<η<-4)-3.5<η<-2.5-2.5<η<-1.5
1-1.5<η<-0.5-0.5<η< 0.5

0-1-1 w.r.t Ψ F (4<η<6)

0 0.5

1 1.5 22.5

3

∆ φ [rad]3∆ φ [rad]0.2<kT<1[GeV/c]

2

0

-2

2.5

-3.5<η<-2.5-2.5<η<-1.5-1.5<η<-0.5-0.5<η< 0.5-2
0 0.5 1 1.5 2 2.5 3∆ φ [rad]22
Rµ2 = Rµ,0
+ 2Rµ,2cos(2Selected events with (Ψ2B-Ψ2F)>0.6

2
2
Ros= 2Ros,2
sin(2

+ ↵)

+ ↵)

Phase diﬀerence between Ψ2B and Ψ2F can be seen in Rs, Ro, and Ros
24

14

R2long [fm2]R2out [fm2]R2side [fm2]BB((FF))  
HHBBTT  rraaddiiii  ww..rr..tt  Ψ22
((η>>00))
10

12

8

AMPT Pb+Pb 2.76TeV

10

10
6
8
5

4
6 0 0.5 1 1.5 2 2.5 3∆ φ [rad]

0 0.5

1 1.5 2

2.5

3

0 0.5

∆ φ [rad]1 1.5 22.5

3

∆ φ [rad]R2os=0

4

2

0
0 0.5

1 1.5 22.5

3

∆ φ [rad]2

R2sl [fm2]R2ol [fm2]R2os [fm2]0.2<kT<1 [GeV/c]

1

2

w.r.t Ψ B (-6<η<-4)-0.5<η< 0.5
0.5<η< 1.5
1
1.5<η< 2.5
2.5<η< 3.5

0

0

-1

-1 w.r.t Ψ F (4<η<6)

-2

0 0.5

1 1.5 22.5

3

∆ φ [rad]-0.5<η< 0.5
0.5<η< 1.5
1.5<η< 2.5
2.5<η< 3.5-2
0 0.5 1 1.5 2 2.5 3∆ φ [rad]Selected events with (Ψ2B-Ψ2F)>0.6
Phase diﬀerence between Ψ2B and Ψ2F can be seen in Rs, Ro, and Ros

25

HBT Interferometry! 1956, H. Brown and R. Twiss, measured angular diameter of Sirius_! 1960, Goldhaber et al., correlation among identical pions in p+p! By quantum interference between two identical particles
wave function for
2 bosons(fermions) :

1
12 = p [ (x1 , p1 ) (x2 , p2 ) ±2(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()
2
2R

detectorp2

q=p1-p2 [GeV/c]


26

Correlation Function
! Experimental Correlation Function C2 is defined as:" R(q): Real pairs at the same event.
" M(q): Mixed pairs selected from different events.
Event mixing was performed using eventswith similar z-vertex, centrality, E.P.
107

R(q)
C2 =
M (q)
q = p1

relative momentum dist.

R(q)

106

M(q)

5

10

104

p2

0

" Real pairs include HBT effects, Coulombinteraction and detector inefficient effect.
Mixed pairs doesn’t include HBT and
Coulomb effects.


0.1

0.15

0.2

0.25

0.3

C2=R/M

1.2

HBT effect1.1

1Coulomb repulsion
0.9
0

27

0.05

0.05

0.1

0.15

0.2

0.25

qinv 0.3
qinv[GeV/c]

3D-HBT Analysis
! Core-Halo picture with “Out-Side-Long” frame"Longitudinal center of mass system (pz1=pz2)"taking into account long lived decay particles=[ (1 + G)Fcoul ] + [1Ro2 qo2

Rs

Rl

beam

C2 =C2core + C2halo
G =exp( Rs2 qs2~kT = 1 (~pT 1 + p~T 2 )2~qo k ~kT , ~qs ? ~kTRo]
Rl2 ql22
2Ros
qs qo )

D

Fcoul: Coulomb correction factor
λ : fraction of pairs in the core

r
o
t
c
e
t
e



   
 
  
    
  


28

Correction of Event Plane Resolution
! Smearing effect by finite resolution of the event plane
Reaction Planetrue sizeEvent Planemeasured sizeReaction Plane! Correction for q-distribution Acrr (q,Ro2 [fm2 ]Rs2 [fm2 ]# model-independent correction"Checked by MC-simulation
25
20

Smeared
Corrected!

j)+2⌃⇣n,m [Ac cos(n j ) + As sin(n j )]n /2⇣n,m =
sin(n /2)hcos(n( mreal ))i"PRC.66, 044903(2002)30j ) =Auncrr (q,event plane resolution

30

w.r.t RP
w.r.t RP

25
20

Uncorrected w.r.t EP
15

15

10

10
0

0.5

1

1.5

2

2.5

3

Uncorrected w.r.t EP
Corrected w.r.t EP
0

Δφ [rad]

29

0.5

1

1.5

2

2.5

3

Δφ [rad]indicates larger apparent source sizes for more central
collisions. We verified that the 0th-order FC corresponds
to the HBT radii from an azimuthally integrated analysis.
Strong 2nd-order oscillations are observed for R2o , R2s ,2
and Rndos , and the signs of the oscillations are qualitatively
self-consistent [10,24], though the amplitude for mostcentral events is small. Similar oscillations were observed
STAR, PRL93, 012301

the
stat
ing
ing

Azimuthal sensitive HBT w.r.t 2 -order event planecentralRout-of-plane

0-5%

10-20%

40-80%

2

30

20

10

in-plane

10

peripheral

2

2

30

2
Rs,2
ε final = 2 2Rs,00.15<kT<0.6 GeV/c

!

2

R = R (Δφ )cos(nΔφ )

20

-2
0

PRC70, 044907 (2004), Blast-wave model
2
s,n

0

R2os (fm )R2l (fm )Rin-planeReaction plane  2nd-order event plane(v2 plane)

2
s,n

2

15

20

Rs2 (fm )Ro2 (fm )

25

π/2

π

0

Φ (radians)

π/2

πFIG. 1 (color online). Squared HBT radii using Eq. (1) relaRs,2 istivesensitive
to final
to the reaction planeangle foreccentricity
three centrality classes. The
solid lines show allowed [24] fits to the individual oscillations.

"Oscillation indicates elliptical shape012301-4to out-of-plane direction.
extended

Results of HBT w.r.t 2nd- and 3rd-order event planes are presented today!
30



FIG
reac
cent
indi

Event Plane Determination
1<|η|<2.8

24 scintillator segments  
φi Ψ3Ψ2Ψ4

<cos(n[ Ψn - Ψr ])>Resolution of Event planes

1

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

0.8

beam axis

! Determined by anisotropic flow itself
using Reaction Plane Detector

1
1
=
tan
n
n

0.6

0.4

⌃wi cos(n i )⌃wi sin(n i )◆! EP resolutions <cos[n(Ψn-Ψreal)]>

0.2"Res{Ψ2} ~ 0.75, Res{Ψ3} ~ 0.34

0
0

✓

10

20

30

40

50

60
70
Centrality [%]31