Physics Le e s B 773 (2017) 68–80
Con en s lis s a ailable a ScienceDi ec
Physics Le e s B
www.else ie .com/loca e/physle b
Linea and non-linea flow mode in Pb–Pb collisions a
√sNN =2.76 TeV
.ALICE Collabo a ion
a i c l e i n o a b s a c
A icle his o y:
Recei ed 22 May 2017
Recei ed in e ised o m 10 July 2017
Accep ed 27 July 2017
A ailable online 4 Augus 2017
Edi o : L. Rolandi
The second and he hi d o de aniso opic flow, V2and V3, a e mos ly de e mined by he co esponding
ini ial spa ial aniso opy coefficien s, ε2and ε3, in he ini ial densi y dis ibu ion. In addi ion o hei
dependence on he same o de ini ial aniso opy coefficien , highe o de aniso opic flow, Vn(n >3),
can also ha e a significan con ibu ion om lowe o de ini ial aniso opy coefficien s, which leads
o mode-coupling e ec s. In his Le e we in es iga e he linea and non-linea modes in highe
o de aniso opic flow Vn o n =4, 5, 6 wi h he ALICE de ec o a he La ge Had on Collide . The
measu emen s a e done o pa icles in he pseudo apidi y ange |η| <0.8and he ans e se momen um
ange 0.2 <pT<5.0GeV/cas a unc ion o collision cen ali y. The esul s a e compa ed wi h heo e ical
calcula ions and p o ide impo an cons ain s on he ini ial condi ions, including ini ial spa ial geome y
and i s fluc ua ions, as well as he a io o he shea iscosi y o en opy densi y o he p oduced sys em.
©2017 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license
(h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by SCOAP3.
1. In oduc ion
The p ima y goal o he ul a- ela i is ic hea y-ion collision
p og amme a he La ge Had on Collide (LHC) is o s udy he
p ope ies o he Qua k–Gluon Plasma (QGP), ano el s a e o
s ongly in e ac ing ma e ha is p oposed o exis a high em-
pe a u es and ene gy densi ies [1,2]. S udies o azimu hal co e-
la ions o p oduced pa icles ha e con ibu ed significan ly o he
cha ac e isa ion o he ma e c ea ed in hea y-ion collisions [3,
4]. Aniso opic flow, which quan ifies he aniso opy o he mo-
men um dis ibu ion o final s a e pa icles, is sensi i e o he
e en -by-e en fluc ua ing ini ial geome y o he o e lap egion,
oge he wi h he anspo p ope ies and equa ion o s a e o he
sys em [4–7]. The success ul desc ip ion o aniso opic flow esul s
by hyd odynamic calcula ions sugges s ha he c ea ed medium
beha es as a nea ly pe ec fluid [4,5] wi h a shea iscosi y o
en opy densi y a io, η/s, close o a conjec u ed lowe bound
1/4π[8]. Aniso opic flow is cha ac e ised using a Fou ie decom-
posi ion o he pa icle azimu hal dis ibu ion in he plane ans-
e se o he beam di ec ion [9,10]:
dN
dϕ∝1+2∞
n=1
ncos[n(ϕ−n)],(1)
whe e Nis he numbe o p oduced pa icles, ϕis he azimu hal
angle o he pa icle and nis he n h o de flow symme y plane.
E-mail add ess: [email p o ec ed].
The n h o de (complex) aniso opic flow Vnis defined as: Vn≡
neinn, whe e n=|Vn|is he flow coefficien , and n ep esen s
he azimu h o Vnin momen um space. Fo non-cen al hea y-ion
collisions, he dominan flow coefficien is 2, e e ed o as ellip ic
flow. Non- anishing alues o highe flow coefficien s 3– 6a he
LHC a e asc ibed p ima ily o he esponse o he p oduced QGP
o fluc ua ions o he ini ial ene gy densi y p ofile o he colliding
nucleons [11–15].
The s anda d (momen -defined) ini ial aniso opy coefficien s
εn oge he wi h hei co esponding ini ial symme y planes (also
called pa icipan planes) ncan be calcula ed om he ans e se
posi ions ( , φ) o he pa icipa ing nucleons
εneinn≡− neinφ
n( o n>1), (2)
whe e deno es he a e age o e he ans e se posi ion o all
pa icipa ing nucleons, φis azimu hal angle, and nis he o de
o he coefficien [11,16]. I has been shown in [17,18] ha V2
and V3a e mos ly de e mined wi h he same o de ini ial spa-
ial aniso opy coefficien s ε2and ε3, espec i ely. Conside ing ha
η/s educes he hyd odynamic esponse o n o εn, i was p o-
posed in [18–21] ha n/εn( o n =2, 3) could be a di ec p obe
o quan i a i ely cons ain he η/so he QGP in hyd odynamic
calcula ions. Howe e , εncanno be de e mined expe imen ally. In-
s ead, hey a e ob ained om a ious heo e ical models, esul ing
in la ge unce ain ies in he es ima ed η/sde i ed indi ec ly om
2and 3measu emen s [17,19]. On he o he hand, highe o de
aniso opic flow Vnwi h n >3p obe smalle spa ial scales and
h p://dx.doi.o g/10.1016/j.physle b.2017.07.060
0370-2693/©2017 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by
SCOAP3.
ALICE Collabo a ion / Physics Le e s B 773 (2017) 68–80 69
hus a e mo e sensi i e o η/s han V2and V3due o mo e p o-
nounced iscous co ec ions [16,22]. Thus, he s udy o he ull se
o flow coefficien s is expec ed o cons ain bo h εnand η/ssi-
mul aneously. Howe e , i was ealised la e ha Vnwi h n >3is
no linea ly co ela ed wi h he co esponding εn[16,22,23], which
makes he ex ac ion o η/s om measu emen s o highe o de
flow coefficien s less s aigh o wa d. In addi ion o he s udy o
flow coefficien s, he esul s o co ela ions be ween di e en o -
de aniso opic flow angles and ampli udes shed ligh on bo h
he ea ly s age dynamics and he anspo p ope ies o he c e-
a ed QGP [24–32]. In pa icula , he cha ac e is ic pa e n o flow
symme y plane co ela ions (also known as angula co ela ions
o flow- ec o s) obse ed in expe imen s is ep oduced quan i a-
i ely by heo e ical calcula ions [29–33]. Howe e , he co ela ions
be ween flow coefficien s (also known as ampli ude co ela ions
o flow- ec o s), in es iga ed using symme ic cumulan s, p o ide
s ic e cons ain s on ini ial condi ions and η/s han he indi id-
ual nmeasu emen s [24–28,31,32]. I is a challenge o cu en
heo e ical models o p o ide quan i a i e desc ip ions o he co -
ela ions be ween di e en o de flow coefficien s.
As discussed abo e, i is known ha he lowe o de aniso opic
flow Vn(n =2, 3) is la gely de e mined by a linea esponse o he
sys em o he co esponding εn(excep in pe iphe al collisions).
Highe o de aniso opic flow Vnwi h n >3 ha e con ibu ions
no only om he linea esponse o he sys em o εn, bu also con-
ibu ions p opo ional o he p oduc o ε2and/o ε3. These con-
ibu ions a e usually called non-linea esponse [25,34] in highe
o de aniso opic flow. Fo a single e en , Vnwi h n =4, 5 and
6 can be decomposed in o he so-called linea and he non-linea
con ibu ions, acco ding o
V4=VNL
4+VL
4=χ4,22(V2)2+VL
4,(3)
V5=VNL
5+VL
5=χ5,32V2V3+VL
5,(4)
V6=VNL
6+VL
6
=χ6,222(V2)3+χ6,33(V3)2+χ6,42V2VL
4+VL
6.(5)
He e χn,mk is a new obse able called he non-linea mode co-
efficien [34] and VNL
n ep esen s he non-linea mode which has
con ibu ions om modes wi h lowe o de aniso opy coefficien s.
The VL
n e m ep esen s he linea mode, which was naï ely ex-
pec ed om he linea esponse o he sys em o he same o -
de εn. Howe e , a ecen hyd odynamic calcula ion showed ha
VL
nis no d i en by he linea esponse o he s anda dly momen -
defined εnin oduced in Eq. (2), bu he co esponding cumulan -
defined aniso opy coefficien ε
n[30,35]. Fo example, VL
4is ex-
pec ed o be d i en by he 4 h-o de cumulan -defined aniso opy
coefficien and i s co esponding ini ial symme y plane which can
be calcula ed as
ε
4ei4
4≡−z4−3z22
4=ε4ei44+3 22
4ε2
2ei42,(6)
whe e z= eiφ. The calcula ions o o he o de aniso opy coe -
ficien s and hei co esponding ini ial symme y planes can be
ound in [30,35]. I he non-linea and linea modes o highe
o de aniso opic flow, VNL
nand VL
n, a e unco ela ed (e.g. VL
nis
pe pendicula o VNL
n), hey can be isola ed. One o he p oposed
app oaches o alida e he assump ion ha VNL
nand VL
na e unco -
ela ed is es ing he ollowing ela ions [25]:
V4(V∗
2)2 2
2
V4(V∗
2)2 2
2= 6
2
4
2 2
2,(7)
V5V∗
3V∗
2 2
2
V5V∗
3V∗
2 2
2= 4
2 2
3
2
2 2
3 2
2.(8)
I he abo e ela ions a e alid, one could combine he analyses
o highe o de aniso opic flow wi h espec o hei co espond-
ing symme y planes and o he planes o lowe o de aniso opic
flow V2o V3 o elimina e he unce ain y om ini ial s a e as-
sump ions and ex ac η/swi h be e p ecision [34].
In his Le e , he linea and non-linea modes in highe o -
de aniso opic flow gene a ion a e s udied in Pb–Pb collisions a
√sNN =2.76 TeV wi h he ALICE de ec o . The main obse ables
a e in oduced in Sec ion 2and he expe imen al se up is de-
sc ibed in Sec ion 3. Sec ion 4p esen s he s udy o he sys ema ic
unce ain ies o he abo e men ioned obse ables. The esul s and
hei discussion a e p o ided in Sec ion 5. Sec ion 6con ains he
summa y and conclusions.
2. Obse ables and analysis me hods
Ideally, he flow coefficien ncan be measu ed ia he az-
imu hal co ela ions o emi ed pa icles wi h espec o he sym-
me y plane nas n=cosn(ϕ−n). Since nis unknown ex-
pe imen ally, he simples app oach o ob ain nis using 2-pa icle
co ela ions:
n{2}=cosn(ϕ1−ϕ2)1/2= 2
n1/2
.(9)
He e deno es he a e age o e all pa icles in a single e en
and hen an a e age o o e all e en s, indica es he e en a e -
age o o e all e en s, and ϕi ep esen s he azimu hal angle o he
i- h pa icle. The analysed e en s a e di ided in o wo sub-e en s
A and B, sepa a ed by a pseudo apidi y gap, o supp ess non-flow
e ec s. The la e a e he azimu hal co ela ions no associa ed o
he common symme y plane n, such as je s and esonance de-
cays. Thus, we modi y Eq. (9) o
n{2}=cos(nϕA
1−nϕB
2)1/2= 2
n1/2
.(10)
He e ϕA
1and ϕB
2a e selec ed om sube en A and B, espec i ely.
Be o e in oducing obse ables ela ed o he linea and non-
linea modes in highe o de aniso opic flow, i is c ucial o e i y
whe he Eqs. (7)–(8) a e applicable. The le and igh hand sides
o Eq. (7) a e ob ained by cons uc ing sui able mul i-pa icle co -
ela ions [34]:
V4(V∗
2)2 2
2A
V4(V∗
2)2A 2
2
=cos(4ϕA
1+2ϕA
2−2ϕB
3−2ϕB
4−2ϕB
5)
cos(4ϕA
1−2ϕB
2−2ϕB
3)cos(2ϕA
1−2ϕB
2),(11)
6
2
4
2 2
2
=cos(2ϕA
1+2ϕA
2+2ϕA
3−2ϕB
4−2ϕB
5−2ϕB
6)
cos(2ϕA
1+2ϕA
2−2ϕB
3−2ϕB
4)cos(2ϕA
1−2ϕB
2).(12)
Simila ly, we can alida e Eq. (8) by calcula ing bo h sides
wi h [34]:
V5V∗
3V∗
2 2
2A
V5V∗
3V∗
2A 2
2
=cos(5ϕA
1+2ϕA
2−3ϕB
3−2ϕB
4−2ϕB
5)
cos(5ϕA
1−3ϕB
2−2ϕB
3)cos(2ϕA
1−2ϕB
2),(13)
70 ALICE Collabo a ion / Physics Le e s B 773 (2017) 68–80
4
2 2
3
2
2 2
3 2
2
=cos(3ϕA
1+2ϕA
2+2ϕA
3−3ϕB
4−2ϕB
5−2ϕB
6)
cos(3ϕA
1+2ϕA
2−3ϕB
3−2ϕB
4)cos(2ϕA
1−2ϕB
2).(14)
The magni ude o VNL
nwas deno ed as n{m}(he e mis he
lowe o de flow symme y plane and m =2, 3) in [34]. The no-
a ion n,mk, whe e nspecifies he o de o he flow e m while
mand ke c. deno e he con ibu ing lowe o de flow symme y
planes, is used in his Le e . I he linea and non-linea modes a e
independen , hen he non-linea mode in highe o de aniso opic
flow can be analysed by co ela ing Vnwi h 2o /and 3[34].
Fo sub-e en A we can define:
A
4,22 =cos(4ϕA
1−2ϕB
2−2ϕB
3)
cos(2ϕA
1+2ϕA
2−2ϕB
3−2ϕB
4)
,(15)
A
5,32 =cos(5ϕA
1−3ϕB
2−2ϕB
3)
cos(3ϕA
1+2ϕA
2−3ϕB
3−2ϕB
4)
,(16)
A
6,222 =cos(6ϕA
1−2ϕB
2−2ϕB
3−2ϕB
4)
cos(2ϕA
1+2ϕA
2+2ϕA
3−2ϕB
4−2ϕB
5−2ϕB
6)
,(17)
A
6,33 =cos(6ϕA
1−3ϕB
2−3ϕB
3)
cos(3ϕA
1+3ϕA
2−3ϕB
3−3ϕB
4)
.(18)
Simila ly, one can ob ain B
n,mk o sub-e en B. The a e age o
A
n,mk and B
n,mk, defined as n,mk, quan ifies he magni ude o he
non-linea mode in highe o de aniso opic flow, which can be
w i en as [36]:
4,22 = 4 2
2cos(44−42)
4
2≈ 4cos(44−42),(19)
5,32 = 5 3 2cos(55−33−22)
2
3 2
2
≈ 5cos(55−33−22),(20)
6,222 = 6 3
2cos(66−62)
6
2≈ 6cos(66−62),(21)
6,33 = 6 2
3cos(66−63)
4
3≈ 6cos(66−63).(22)
The app oxima ion is alid i he co ela ion be ween lowe (n =
2, 3) and highe (n >3) flow coefficien s is weak.
As can be seen in Eqs. (3)–(5), he calcula ion o V6is mo e
complica ed han V4and V5, and he exac exp ession o L
6is
cu en ly no a ailable. The e o e, we only ocus on he wo non-
linea modes o V6wi hou discussing L
6. Acco ding o Eqs. (3) o
(4), he magni udes o he linea mode in highe o de aniso opic
flow can be calcula ed as:
L
4= 2
4{2}− 2
4,22,(23)
L
5= 2
5{2}− 2
5,32.(24)
The a io o n,mk o n{2}, deno ed as ρn,mk, can be calcula ed
as:
ρ4,22 = 4,22
4{2}=cos(44−42),(25)
ρ5,32 = 5,32
5{2}=cos(55−33−22),(26)
ρ6,222 = 6,222
6{2}=cos(66−62),(27)
ρ6,33 = 6,33
6{2}=cos(66−63).(28)
These obse ables measu e he co ela ions be ween di e en o -
de flow symme y planes i he co ela ions be ween di e en
o de flow coefficien s a e weak. They a e e y simila o he so-
called weigh ed e en -plane co ela ions measu ed by he ATLAS
Collabo a ion [33]. The di e ences a e as ollows: 2
2 2
3is used in
Eq. (20) and (26), while 2
2 2
3was used in [33], which did no
conside he an i-co ela ions be ween 2and 3 ound in [27]. In
addi ion, mul i-pa icle co ela ions a e used o 2and 3in he
denomina o o he obse ables, while wo-pa icle co ela ions a e
used in he e en -plane co ela ions which migh be biased om
fluc ua ions o 2and 3.
The non-linea mode coefficien s χn,mk in Eqs. (3) o (5) a e
defined as:
χ4,22 = 4,22
4
2
(29)
χ5,32 = 5,32
2
2 2
3
(30)
χ6,222 = 6,222
6
2
(31)
χ6,33 = 6,33
4
3
.(32)
These quan i y he con ibu ions o he non-linea mode and a e
expec ed o be independen o 2o 3.
All o he obse ables abo e a e based on 2- and mul i-pa icle
co ela ions, which can be ob ained using he gene ic amewo k
o aniso opic flow analyses in oduced in Re . [24].
3. Expe imen al se up and da a analysis
The da a samples analysed in his Le e we e eco ded by
ALICE du ing he Pb–Pb uns o he LHC a a cen e-o -mass en-
e gy o √sNN =2.76 TeV in 2010. Minimum bias Pb–Pb collision
e en s we e igge ed by he coincidence o signals in he V0 de-
ec o [37,38], wi h an efficiency o 98.4% o he had onic c oss
sec ion [39]. The V0 de ec o is composed o wo a ays o scin-
illa o coun e s, V0-A and V0-C, which co e he pseudo apidi y
anges 2.8 <η<5.1 and −3.7 <η<−1.7, espec i ely. Beam
backg ound e en s we e ejec ed using he iming in o ma ion
om he V0 and he Ze o Deg ee Calo ime e (ZDC) [37] de ec o s
and by co ela ing econs uc ed clus e s and ackle s wi h he Sil-
icon Pixel De ec o s (SPD). The ac ion o pile-up e en s in he
da a sample is ound o be negligible a e applying dedica ed pile-
up emo al c i e ia [40]. Only e en s wi h a econs uc ed p ima y
e ex wi hin ±10 cm om he nominal in e ac ion poin along
he beam di ec ion we e used in his analysis. The p ima y e -
ex was es ima ed using acks econs uc ed by he Inne T acking
Sys em (ITS) [37,41] and Time P ojec ion Chambe (TPC) [37,42].
The collision cen ali y was de e mined om he measu ed V0
ampli ude and cen ali y in e als we e defined ollowing he p o-
cedu e desc ibed in [39]. Abou 13 million Pb–Pb e en s passed all
o he e en selec ion c i e ia.
ALICE Collabo a ion / Physics Le e s B 773 (2017) 68–80 71
T acks econs uc ed using he combined in o ma ion om he
TPC and ITS a e used in his analysis. This combina ion ensu es
a high de ec ion efficiency, op imum momen um esolu ion, and
a minimum con ibu ion om pho on con e sions and seconda y
cha ged pa icles p oduced ei he in he de ec o ma e ial o
om weak decays. To educe he con ibu ions om seconda ies,
cha ged acks we e equi ed o ha e a dis ance o closes ap-
p oach o he p ima y e ex in he longi udinal (z) di ec ion and
ans e se (xy) plane smalle han 3.2 cm and 2.4 cm, espec-
i ely. Addi ionally, acks we e equi ed o ha e a leas 70 TPC
space poin s ou o he maximum 159. The a e age χ2pe deg ee
o eedom o he ack fi o he TPC space poin s was equi ed
o be below 2. In his s udy, acks we e selec ed in he pseu-
do apidi y ange |η| <0.8 and he ans e se momen um ange
0.2 <pT<5.0GeV/c.
4. Sys ema ic unce ain ies
Nume ous sou ces o sys ema ic unce ain y we e in es iga ed
by a ying he e en and ack selec ion as well as he unce ain y
associa ed wi h he possible emaining non-flow e ec s in he
analysis. The a ia ion o he esul s wi h he collision cen ali y
is calcula ed by al e na i ely using he TPC o SPD o es ima e he
e en mul iplici y and is ound o be less han 3% o all obse -
ables. Resul s wi h opposi e pola i ies o he magne ic field wi hin
he ALICE de ec o and wi h na owing he nominal ±10 cm ange
o he econs uc ed e ex along he beam di ec ion om he cen-
e o he ALICE de ec o o 9, 8 and 7 cm do no show a di e ence
o mo e han 2% compa ed o he de aul selec ion c i e ia o a -
ious measu emen s. The con ibu ions om pile-up e en s o he
final sys ema ic unce ain y a e ound o be negligible. The sensi-
i i y o he ack selec ion c i e ia was explo ed by a ying he
numbe o TPC space poin s and by using acks econs uc ed in
he TPC alone. Va ying he numbe o TPC space poin s om 70
o 80, 90 and 100 ou o a possible 158, esul s in a 1–3% a ia-
ion o he esul s o n, wi hin 1.5% o ρn,mk and χn,mk. Using
TPC-only acks leads o a di e ence o less han 14%, 17% and 8%
o n, ρn,mk and χn,mk, espec i ely. Bo h e ec s we e included in
he e alua ion o he sys ema ic unce ain y. Se e al di e en ap-
p oaches ha e been applied o es ima e he e ec s o non-flow.
These include he in es iga ion o mul i-pa icle co ela ions wi h
a ious |η|gaps, he applica ion o he like-sign echnique which
co ela es wo pa icles wi h ei he all posi i e o nega i e cha ges
and supp ess such non-flow as due o esonance decays, as well as
he calcula ions using HIJING Mon e Ca lo simula ions [43], which
do no include aniso opic flow. I was ound ha he possible e-
maining non-flow e ec s a e less han 10.5%, 11% and 7% o n,
ρn,mk and χn,mk, espec i ely. They a e aken in o accoun in he fi-
nal sys ema ic unce ain y. The sys ema ic unce ain ies e alua ed
o each sou ce men ioned abo e we e added in quad a u e o ob-
ain he o al sys ema ic unce ain y o he measu emen s.
5. Resul s and discussion
As discussed in Sec. 2, one can alida e he assump ion ha
linea and non-linea modes in highe o de aniso opic flow a e
unco ela ed ia Eqs. (7) and (8). These ha e been es ed in A
Mul i-Phase T anspo (AMPT) model [25] as well as in he hy-
d odynamic calcula ions [44]. Good ag eemen be ween le - and
igh -hand sides o Eqs. (7) and (8) is ound o all cen ali y
classes, independen o he ini ial condi ions and he ideal o is-
cous fluid dynamics used in he calcula ions. Thus, i is c ucial
o check hese equali ies in da a, o u he confi m he assump-
ion ha he wo componen s a e unco ela ed and can be iso-
la ed independen ly. Fig. 1 confi ms ha he ag eemen obse ed
Fig. 1. S udy o ela ionship be ween linea and non-linea modes in highe o -
de aniso opic flow in Pb–Pb collisions a √sNN =2.76 TeV, acco ding o Eqs. (7)
and (8).
in heo e ical calcula ions is also p esen in he da a despi e small
de ia ions ound in cen al collisions when es ing Eqs. (8). Thei
cen ali y dependency a e simila as he p e ious heo e ical p e-
dic ions [25,44]. The measu emen s suppo he assump ion ha
highe o de aniso opic flow Vn(n >3) can be modeled as he
sum o independen linea and non-linea modes.
The magni udes o linea and non-linea modes in highe o de
aniso opic flow a e epo ed as a unc ion o collision cen al-
i y in Fig. 2. In his Le e , sub-e en s A and B a e buil in he
pseudo apidi y anges −0.8 <η<−0.4 and 0.4 <η<0.8, e-
spec i ely, which esul s in a pseudo apidi y gap o |η| >0.8 o
all p esen ed measu emen s. I can be seen ha he linea mode
L
4depends weakly on cen ali y and is he la ge con ibu ion o
4{2} o he cen ali y ange 0–30%. The non-linea mode, 4,22,
inc eases mono onically as he cen ali y dec eases and sa u a es
a ound cen ali y pe cen ile 50%, becoming he dominan sou ce
o cen ali y in e als abo e 40%. Simila ends o cen ali y de-
pendence ha e been obse ed o V5, al hough 5,32 becomes he
dominan con ibu ion in cen ali y pe cen ile abo e 30%. Only wo
non-linea componen s 6,222 and 6,33 a e discussed o V6. I is
shown in Fig. 2 ( igh ) ha 6,222 inc eases mono onically as he
cen ali y dec eases o cen ali y 50%, while 6,33 has a weake
cen ali y dependence compa ed o 6,222.
The linea and non-linea modes in highe o de aniso opic
flow we e in es iga ed by he ATLAS Collabo a ion [26] using a
di e en app oach based on “E en Shape Enginee ing” [45]. Wi h
his me hod one can u ilise la ge fluc ua ions in he ini ial geome-
y o he sys em o selec e en s co esponding o a specific ini ial
shape. The conclusion is quali a i ely consis en wi h wha is e-
po ed he e, al hough a di ec compa ison is no possible due o
he di e en kinema ic cu s (especially he in eg a ed pT ange)
used in he wo measu emen s. The highe o de aniso opic flow
induced by lowe o de aniso opic flow we e also measu ed us-
ing he e en -plane me hod a he LHC [14,46]. Howe e , he
measu emen s o he non-linea mode p esen ed in his Le e
a e based on he mul i-pa icle co ela ions me hod wi h a |η|
gap. This me hod makes i easie o measu e an obse able like
5,32, which is less s aigh o wa d o define using he e en plane
me hod [14,46]. In addi ion, as poin ed ou in [25,34,47], his
new mul i-pa icle co ela ions me hod should s ongly supp ess
sho - ange (in pseudo apidi y) non-flow e ec s and p o ides a o-
bus measu emen wi hou any dependence on he expe imen al
accep ance. The measu emen s a e compa ed o ecen hyd ody-
namic calcula ions om a hyb id IP-Glasma +MUSIC +U QMD
model [48], in which ealis ic e en -by-e en ini ial condi ions a e
used and he hyd odynamic e olu ion akes in o accoun bo h
shea and bulk iscosi y. I is shown ha his hyd odynamic cal-
cula ion could desc ibe quan i a i ely he o al magni udes o V4
and V6, as well as he magni udes o hei linea and non-linea
modes, while i sligh ly o e es ima es he esul s o V5.
The cen ali y dependence o ρn,mk, which quan ifies he angu-
la co ela ions be ween di e en o de flow symme y planes, is
72 ALICE Collabo a ion / Physics Le e s B 773 (2017) 68–80
Fig. 2. Cen ali y dependence o 4(le ), 5(middle) and 6( igh ) in Pb–Pb collisions a
√sNN =2.76 TeV. Con ibu ions om linea and non-linea modes a e p esen ed
wi h open and solid ma ke s, espec i ely. The hyd odynamic calcula ions om IP-Glasma +MUSIC +U QMD [48] a e shown o compa ison.
Fig. 3. Cen ali y dependence o ρn,mk in Pb–Pb collisions a
√sNN =2.76 TeV. ATLAS
measu emen s based on he e en -plane co ela ion [33] a e p esen ed wi h open
ma ke s. The hyd odynamic calcula ions om IP-Glasma +MUSIC +U QMD [48]
a e shown wi h open bands. (Fo in e p e a ion o he e e ences o colou in his
figu e legend, he eade is e e ed o he web e sion o his a icle.)
p esen ed in Fig. 3. I is obse ed ha ρ4,22 inc eases om cen-
al o pe iphe al collisions, which sugges s ha he co ela ions
be ween 2and 4a e s onge in pe iphe al han in cen al
collisions. I implies ha VNL
4 ends o align wi h V4in mo e
pe iphe al collisions. The esul s o ρ6,33, which measu es he
co ela ion o 3and 6, do no exhibi a s ong cen ali y de-
pendence wi hin he s a is ical unce ain ies. As men ioned abo e,
ρ4,22 and ρ6,33 a e simila o he p e ious “e en -plane co ela-
ion” measu emen s cos(44−42)wand cos(66−63)w
in [33]. The compa isons be ween measu emen s o hese ob-
se ables a e also p esen ed in Fig. 3. The esul s a e compa ible
wi h each o he , despi e he di e en kinema ic anges used by
ATLAS and his analysis. I should be also no ed ha he mea-
su emen s o ρn,mk p esen ed in his Le e show he symme y
plane co ela ions a mid-pseudo apidi y |η| <0.8 while ATLAS
measu ed he symme y plane co ela ions using −4.8 <η<−0.5
and 0.5 <η<4.8 o wo-plane co ela ions, and using −2.7 <
η<−0.5, 0.5 <η<2.7 and 3.3 <|η| <4.8 o 3-plane co e-
la ions [33]. P e ious in es iga ions sugges ha he e migh be
η-dependen fluc ua ions o he flow symme y plane and he flow
magni ude [49,50]. As a consequence, one migh expec a di e -
ence when measu ing he co ela ions o flow symme y planes
om di e en pseudo apidi y egions. Howe e , Fig. 3 shows good
ag eemen be ween he ALICE and ATLAS measu emen s. The e-
o e, no ob ious indica ion ha he flow symme y plane a ies
wi h ηcan be deduced om his compa ison. I is no iceable in
Fig. 3 ha he ρ5,32 measu emen seems sligh ly highe han he
cos(55−33−22)wmeasu emen . This is mainly due o a
small di e ence be ween he defini ions o he obse able as in-
oduced in Sec. 2: he e m 2
2 2
31/2is used in ρ5,32, whe eas
2
21/2 2
31/2is used in he “e en -plane co ela ions” [33]. Con-
side ing he known an i-co ela ions be ween 2and 3[26,27],
2
2 2
31/2could be up o 10% lowe han 2
21/2 2
31/2depending
on he cen ali y class [27], leading o a sligh ly la ge ρ5,32 han
cos(55−33−22)w om ATLAS.
I has been obse ed in hyd odynamic and anspo model cal-
cula ions ha he symme y plane co ela ions, e.g. co ela ions
o second and ou h o de symme y planes, change sign du -
ing he sys em e olu ion [29,30,32]. The measu ed flow symme y
plane co ela ions could be nicely explained by he combina ion
o con ibu ions om linea and non-linea modes in highe o -
de aniso opic flow [30]. This indica es ha he flow symme y
plane co ela ion ρn,mk ca ies impo an in o ma ion abou he
dynamic e olu ion o he c ea ed sys em. In addi ion, he model
calcula ions sugges ha s onge ini ial symme y plane co e-
la ions a e eflec ed in s onge co ela ions be ween he flow
symme y planes in he final s a e [29,32]. And a la ge alue
o η/so he QGP leads o weake flow symme y plane co e-
la ions in he final s a e. As poin ed ou in [29], he hyd ody-
namic calcula ions om VISH2 +1using Mon e Ca lo Glaube
(MC-Glb) o Mon e Ca lo Kha zee –Le in–Na di (MC-KLN) ini ial
condi ions can only desc ibe quali a i ely he ends o he cen-
ali y dependence o he e en -plane co ela ion measu emen s
by ATLAS. I is he e o e expec ed ha hese hyd odynamic calcu-
la ions canno desc ibe he p esen ed ALICE measu emen s, which
a e compa ible wi h he ATLAS e en -plane co ela ion measu e-
men s. Fig. 3 shows ha he hyd odynamic calcula ions om
IP-Glasma +MUSIC +U QMD [48] ep oduce nicely he mea-
su emen s o symme y plane co ela ions ρn,mk. The measu e-
men s o ρn,mk p esen ed in his Le e , oge he wi h he com-
pa ison o hyd odynamic calcula ions, should place cons ain s on
he ini ial condi ions and η/so he QGP in hyd odynamic calcula-
ions.
Fig. 4 p esen s he measu emen s o he non-linea mode coe -
ficien s as a unc ion o collision cen ali y. I is obse ed ha χ4,22
and χ6,222 dec ease modes ly om cen al o mid-cen al colli-
sions, and s ay almos cons an om mid-cen al o mo e pe iph-
e al collisions. Fo χ5,32 and χ6,33 s ong cen ali y dependence is
no obse ed ei he . Thus, he d ama ic inc ease o n,mk shown
in Fig. 2 appea s o be mainly due o he inc ease o 2and/o
3 om cen al o pe iphe al collisions and no he inc ease o he
non-linea mode coefficien . I is also no ewo hy ha he ela ion-
ship o χ4,22 ∼χ6,33 ≈χ5,32
2is app oxima ely alid, as p edic ed
by hyd odynamic calcula ions [34]. The compa isons o e en -by-
e en iscous hyd odynamic calcula ions om VISH2 +1[44] and
om IP-Glasma +MUSIC +U QMD [48] a e also p esen ed in
Fig. 4. VISH2 +1shows ha χ4,22 calcula ions wi h MC-Glb ini-
ial condi ions a e la ge han hose wi h MC-KLN ini ial condi-
ions, i.e. χ4,22 depends on he ini ial condi ions. A he same ime,
ALICE Collabo a ion / Physics Le e s B 773 (2017) 68–80 73
Fig. 4. Cen ali y dependence o χin Pb–Pb collisions a
√sNN =2.76 TeV. Hyd o-
dynamic calcula ions om VISH2 +1[44] a e shown in shaded a eas and he one
om IP-Glasma +MUSIC +U QMD [48] a e shown wi h open bands. (Fo in e -
p e a ion o he e e ences o colou in his figu e legend, he eade is e e ed o
he web e sion o his a icle.)
he cu es wi h di e en η/s alues o VISH2 +1a e e y sim-
ila , indica ing ha χ4,22 is insensi i e o η/s. The measu emen s
a ou IP-Glasma and MC-KLN o e MC-Glb ini ial condi ions e-
ga dless o η/s. This sugges s ha he χ4,22 measu emen can be
used o cons ain he ini ial condi ions, wi h less conce n o he
se ing o η/s(T)in hyd odynamic calcula ions han p e ious flow
obse ables.
I was p edic ed ha χ6,222 <χ6,33 based on he ideal hy-
d odynamic calcula ion using smoo h ini ial Gaussian densi y p o-
files [34], whe eas an opposi e p edic ion was ob ained in he
ideal hyd odynamic calcula ion e ol ing genuinely bumpy ini ial
condi ions ob ained om a Mon e Ca lo sampling o he ini ial nu-
cleon posi ions in he colliding nuclei [44]. I is seen in Fig. 4 ha
χ6,222 ∼χ6,33 wi hin he cu en unce ain ies. The da a a e no
able o disc imina e he di e en p edic ions in [34] and [44]. Hy-
d odynamic calcula ions using MC-KLN and IP-Glasma ini ial con-
di ions gi e be e desc ip ions o χ6,222, compa ed o he ones
using MC-Glb ini ial condi ions. Fo χ5,32 none o he combina-
ions o ini ial condi ions and η/sin he hyd odynamic calcula ions
ag ee quan i a i ely wi h da a. This migh be due o he cu en
difficul y o desc ibing he an i-co ela ions be ween 2and 3
in hyd odynamic calcula ions [27,51], which a e in ol ed in he
calcula ion o χ5,32. Fu he mo e, VISH2 +1calcula ions show
ha χ5,32 and χ6,33 a e e y weakly sensi i e o he ini ial con-
di ions, bu dec ease as η/sinc eases. The in es iga ion wi h he
VISH2 +1hyd odynamic amewo k shows ha he sensi i i y
o χ5,32 and χ6,33 o η/sis no due o sensi i i y o shea is-
cous e ec s du ing he buildup o hyd odynamic flow. Ins ead, as
ound in [44], i is due o he η/sa eeze-ou . The measu emen s
o χ5,32 and χ6,33 do no u he cons ain he η/sdu ing sys-
em e olu ion, howe e , hey p o ide unique in o ma ion on η/sa
eeze-ou which was poo ly known and canno be ob ained om
o he aniso opic flow ela ed obse ables. Fu he imp o emen o
model calcula ions on co ela ions be ween di e en o de flow
coefficien s a e necessa y o be e unde s and he compa ison
o χ5,32 ob ained om da a and hyd odynamic calcula ions. The
χ6,33 esul s a e consis en wi h hyd odynamic calcula ions om
VISH2 +1wi h MC-KLN ini ial condi ions using η/s =0.08 and
IP-Glasma +MUSIC +U QMD wi h a η/s =0.095. I is shown
ha χ5,32 and χ6,33 ha e a weak cen ali y dependence i a
smalle η/sis used in he hyd odynamic calcula ions. Such a weak
cen ali y dependence o χ5,32 and χ6,33 is obse ed in da a as
well. The measu emen s p esen ed he e sugges a small η/s alue
a eeze-ou , which can be use ul o cons ain he empe a u e de-
pendence o he shea iscosi y o e en opy densi y a io, η/s(T),
in he de elopmen o hyd odynamic amewo ks. These esul s
sugges ha u u e uning o he pa ame e isa ions o η/s(T)in
hyd odynamic amewo ks using he p esen ed measu emen s is
necessa y.
6. Summa y
The linea and non-linea modes in highe o de aniso opic
flow gene a ion we e s udied wi h 2- and mul i-pa icle co e-
la ions in Pb–Pb collisions a √sNN =2.76 TeV. The esul s p e-
sen ed in his Le e show ha highe o de aniso opic flow can
be isola ed in o wo independen con ibu ions: he componen
ha a ises om a non-linea esponse o he sys em o he lowe
o de ini ial aniso opy coefficien s ε2and/o ε3, and a linea mode
which is d i en by linea esponse o he sys em o he same o de
cumulan -defined aniso opy coefficien . Aweak cen ali y depen-
dence is obse ed o he con ibu ions om linea mode whe eas
he con ibu ions om non-linea mode inc ease d ama ically as
he collision cen ali y dec eases, and i becomes he dominan
sou ce in highe o de aniso opic flow in mid-cen al o pe iph-
e al collisions. I is shown ha his is mainly due o he inc ease
o lowe o de flow coefficien s 2and 3. The co ela ions be-
ween di e en flow symme y planes a e measu ed. The esul s
a e compa ible wi h he p e ious “e en -plane co ela ion” mea-
su emen s, and can be quan i a i ely desc ibed by calcula ions us-
ing he IP-Glasma +MUSIC +U QMD amewo k. Fu he mo e,
non-linea mode coefficien s, which ha e di e en sensi i i ies o
he shea iscosi y o e en opy densi y a io η/sand he ini ial
condi ions, a e p esen ed in his Le e . Compa isons o hyd ody-
namic calcula ions sugges ha he da a is desc ibed be e by
hyd odynamic calcula ions wi h smalle η/s. In addi ion, he MC-
Glb ini ial condi ion is dis a ou ed by he p esen ed esul s.
Measu emen s o linea and non-linea modes in highe o de
aniso opic flow and hei compa ison o hyd odynamic calcula-
ions p o ide mo e p ecise cons ain s on he ini ial condi ions and
empe a u e dependence o η/s. These esul s could also o e new
insigh s in o he geome y o he fluc ua ing ini ial s a e and in o
he dynamical e olu ion o he s ongly in e ac ing medium p o-
duced in ela i is ic hea y-ion collisions a he LHC.
Acknowledgemen s
The ALICE Collabo a ion would like o hank all i s enginee s
and echnicians o hei in aluable con ibu ions o he cons uc-
ion o he expe imen and he CERN accele a o eams o he ou -
s anding pe o mance o he LHC complex. The ALICE Collabo a ion
g a e ully acknowledges he esou ces and suppo p o ided by
all G id cen es and he Wo ldwide LHC Compu ing G id (WLCG)
collabo a ion. The ALICE Collabo a ion acknowledges he ollow-
ing unding agencies o hei suppo in building and unning
he ALICE de ec o : A. I. Alikhanyan Na ional Science Labo a o y
(Ye e an Physics Ins i u e) Founda ion (ANSL), S a e Commi ee o
Science and Wo ld Fede a ion o Scien is s (WFS), A menia; Aus-
ian Academy o Sciences and Ös e eichische Na ionals i ung
ü Fo schung, Technologie und En wicklung, Aus ia; Minis y o
Communica ions and High Technologies, Na ional Nuclea Resea ch
Cen e , Aze baijan; Conselho Nacional de Desen ol imen o Cien í-
fico e Tecnológico (CNPq), Uni e sidade Fede al do Rio G ande do
Sul (UFRGS), Financiado a de Es udos e P oje os (Finep) and Fun-
dação de Ampa o à Pesquisa do Es ado de São Paulo (FAPESP),
74 ALICE Collabo a ion / Physics Le e s B 773 (2017) 68–80
B azil; Minis y o Science & Technology o China (MSTC), Na-
ional Na u al Science Founda ion o China (NSFC) and Minis y
o Educa ion o China (MOEC), China; Minis y o Science, Edu-
ca ion and Spo s and C oa ian Science Founda ion, C oa ia; Min-
is y o Educa ion, You h and Spo s o he Czech Republic, Czech
Republic; The Danish Council o Independen Resea ch — Na -
u al Sciences, he Ca lsbe g Founda ion and Danish Na ional Re-
sea ch Founda ion (DNRF), Denma k; Helsinki Ins i u e o Physics
(HIP), Finland; Commissa ia à l’Ene gie A omique (CEA) and Ins i-
u Na ional de Physique Nucléai e e de Physique des Pa icules
(IN2P3) and Cen e Na ional de la Reche che Scien ifique (CNRS),
F ance; Bundesminis e ium ü Bildung, Wissenscha , Fo schung
und Technologie (BMBF) and GSI Helmhol zzen um ü Schwe-
ionen o schung GmbH, Ge many; Gene al Sec e a ia o Resea ch
and Technology, Minis y o Educa ion, Resea ch and Religions,
G eece; Na ional Resea ch, De elopmen and Inno a ion Office,
Hunga y; Depa men o A omic Ene gy, Go e nmen o India
(DAE) and Council o Scien ific and Indus ial Resea ch (CSIR),
New Delhi, India; Indonesian Ins i u e o Science, Indonesia; Cen-
o Fe mi -Museo S o ico della Fisica e Cen o S udi e Rice che
En ico Fe mi and Is i u o Nazionale di Fisica Nuclea e (INFN), I aly;
Ins i u e o Inno a i e Science and Technology, Nagasaki Ins i u e
o Applied Science (IIST), Japan Socie y o he P omo ion o Sci-
ence (JSPS) KAKENHI and Japanese Minis y o Educa ion, Cul u e,
Spo s, Science and Technology (MEXT), Japan; Consejo Nacional
de Ciencia y Tecnología (CONACYT), h ough Fondo de Coope a ión
In e nacional en Ciencia y Tecnología (FONCICYT) and Di ección
Gene al de Asun os del Pe sonal Académico (DGAPA), Mexico;
Nede landse O ganisa ie oo We enschappelijk Onde zoek (NWO),
Ne he lands; The Resea ch Council o No way, No way; Commis-
sion on Science and Technology o Sus ainable De elopmen in
he Sou h (COMSATS), Pakis an; Pon ificia Uni e sidad Ca ólica del
Pe ú, Pe u; Minis y o Science and Highe Educa ion and Na ional
Science Cen e, Poland; Ko ea Ins i u e o Science and Technol-
ogy In o ma ion and Na ional Resea ch Founda ion o Ko ea (NRF),
Republic o Ko ea; Minis y o Educa ion and Scien ific Resea ch,
Ins i u e o A omic Physics and Romanian Na ional Agency o Sci-
ence, Technology and Inno a ion, Romania; Join Ins i u e o Nu-
clea Resea ch (JINR), Minis y o Educa ion and Science o he Rus-
sian Fede a ion and Na ional Resea ch Cen e Ku cha o Ins i u e,
Russia; Minis y o Educa ion, Science, Resea ch and Spo o he
Slo ak Republic, Slo akia; Na ional Resea ch Founda ion o Sou h
A ica, Sou h A ica; Cen o de Aplicaciones Tecnológicas y Desa -
ollo Nuclea (CEADEN), Cubaene gía, Cuba; Minis e io de Ciencia e
Inno ación and Cen o de In es igaciones Ene gé icas, Medioambi-
en ales y Tecnológicas (CIEMAT), Spain; Swedish Resea ch Council
(VR) and Knu & Alice Wallenbe g Founda ion (KAW), Sweden; Eu-
opean O ganiza ion o Nuclea Resea ch, Swi ze land; Na ional
Science and Technology De elopmen Agency (NSDTA), Su ana ee
Uni e si y o Technology (SUT) and Office o he Highe Educa-
ion Commission unde NRU p ojec o Thailand, Thailand; Tu kish
A omic Ene gy Agency (TAEK), Tu key; Na ional Academy o Sci-
ences o Uk aine, Uk aine; Science and Technology Facili ies Coun-
cil (STFC), Uni ed Kingdom; Na ional Science Founda ion o he
Uni ed S a es o Ame ica (NSF) and Uni ed S a es Depa men o
Ene gy, Office o Nuclea Physics (DOE NP), Uni ed S a es o Ame -
ica.
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