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Cha ge-dependen low and he sea ch o he chi al magne ic wa e in Pb-Pb
collisions a √sNN = 2.76 TeV
ALICE Collabo a ion
ALICE Collabo a ion. (2016). Cha ge-dependen low and he sea ch o he chi al
magne ic wa e in Pb-Pb collisions a √sNN = 2.76 TeV. Physical Re iew C, 93(4),
A icle 044903. h ps://doi.o g/10.1103/PhysRe C.93.044903
2016
PHYSICAL REVIEW C 93, 044903 (2016)
Cha ge-dependen low and he sea ch o he chi al magne ic wa e in
Pb-Pb collisions a √sNN =2.76 TeV
J. Adam e al.∗
(ALICE Collabo a ion)
(Recei ed 18 Decembe 2015; published 8 Ap il 2016)
We epo on measu emen s o a cha ge-dependen low using a no el h ee-pa icle co ela o wi h ALICE in
Pb-Pb collisions a he CERN La ge Had on Collide (LHC), and discuss he implica ions o obse a ion o local
pa i y iola ion and he chi al magne ic wa e (CMW) in hea y-ion collisions. Cha ge-dependen low is epo ed
o di e en collision cen ali ies as a unc ion o he e en cha ge asymme y. While ou esul s a e in quali a i e
ag eemen wi h expec a ions based on he CMW, he nonze o signal obse ed in highe ha monics co ela ions
indica es a possible signi ican backg ound con ibu ion. We also p esen esul s on a di e en ial co ela o , whe e
he low o posi i e and nega i e cha ges is epo ed as a unc ion o he mean cha ge o he pa icles and hei
pseudo apidi y sepa a ion. We a gue ha his di e en ial co ela o is be e sui ed o dis inguish he di e ences
in posi i e and nega i e cha ges expec ed due o he CMW and he backg ound e ec s, such as local cha ge
conse a ion coupled wi h s ong adial and aniso opic low.
DOI: 10.1103/PhysRe C.93.044903
I. INTRODUCTION
Pa i y (P) is a majo symme y o classical physics, being
p esen in igid-body dynamics, classical elec odynamics, and
g a i y. In he de elopmen o quan um mechanics, pa i y
conse a ion was assumed. I was no un il he 1950s [1] ha
he possibili y o pa i y iola ion was conside ed, and soon
a e i was de ini i ely demons a ed expe imen ally in nuclea
decays [2,3]. In he mode n pic u e, P and CP iola ion in weak
in e ac ions a e widely es ablished expe imen ally and well
unde s ood heo e ically. In s ong in e ac ions he e is e y
li le o no global P iola ion, as de e mined by measu emen s
o he neu on elec ic dipole momen [4,5]. Howe e , he e is
no i s -p inciples eason why P and CP iola ion should no
exis in s ong in e ac ions. P and CP iola ion as a gene al
ea u e o quan um ield heo ies was i s explo ed in he
1970s [6,7], and a p oposal o use hea y-ion collisions as a
ool o s udying P and CP iola ion i s appea ed as ea ly
as he 1980s [8]. Speci ic p oposals o a sea ch o local
P- iola ing e ec s in hea y-ion collisions appea ed in he las
decade [9–14].
Collisions o hea y nuclei a ul a ela i is ic ene gies c ea e
a ho , dense medium ha appea s o ha e pa onic deg ees
o eedom and e ol e hyd odynamically. In noncen al
collisions he ini ial o e lap egion is noniso opic, which, due
o pa icle in e ac ions, leads o a momen um-space aniso opy
o he p oduced pa icles. This aniso opy can be desc ibed
using a Fou ie expansion o he azimu hal dis ibu ion o
pa icles [15]. Noncen al collisions a e also cha ac e ized by
la ge o bi al momen um and, o impo ance o his s udy, e y
∗Full au ho lis gi en a he end o he a icle.
Published by he Ame ican Physical Socie y unde he e ms o he
C ea i e Commons A ibu ion 3.0 License. Fu he dis ibu ion o
his wo k mus main ain a ibu ion o he au ho (s) and he published
a icle’s i le, jou nal ci a ion, and DOI.
la ge magne ic ields. Nume ical es ima es [16,17] indica e
ha a CERN La ge Had on Collide (LHC) ene gies he ield
s eng h can be as la ge as B≈m2
π/e ≈1014 T. In a acuum,
he magne ic ield induced by he spec a o s decays in ime
quad a ically (B∝ −2) and he li e ime o he magne ic ield
a LHC ene gies is ex emely sho , dec easing six o de s o
magni ude o e he cou se o 0.5 m/c. Howe e he p esence
o elec ical cha ges (such as he qua ks in he QGP) means
he e is ini e elec ical conduc i i y. By Lenz’s law he change
in magne ic ield is opposed by he cha ge ca ie s in he
conduc o , so he empo al decay o he magne ic ield is
signi ican ly slowed [18,19].
The chi al magne ic e ec (CME) [20] is a p ocess o cha ge
sepa a ion wi h espec o he eac ion plane. In he QCD
acuum he e can exis gluonic con igu a ions wi h nonze o
opological cha ge, which can be an ins an on o a sphale on.
A high empe a u es, he sphale on a e is expec ed o be
dominan . The p esence o such gluonic con igu a ions wi h
opological cha ge is wha d i es he P- iola ion p ocess. Fo
example, in a egion wi h nega i e opological cha ge, le -
handed qua ks will become igh -handed, and igh -handed
qua ks will emain igh -handed. The s ong magne ic ield
c ea ed in hea y-ion collisions in e ac s wi h he magne ic
momen o he qua ks and o ien s he spins o qua ks wi h
posi i e (nega i e) elec ic cha ge o be pa allel (an ipa allel)
o he ield di ec ion. Unde he assump ion o massless qua ks,
igh -handed qua ks ha e hei spins and momen a aligned.
This will cause posi i e (nega i e) qua ks o mo e pa allel
(an ipa allel) o he magne ic ield, leading o a posi i e
elec ic cu en and hus a posi i e elec ic cha ge dipole.
Due o he chi al symme y es o a ion, uand dqua ks
ha e only hei ba e Higgs mass, which is o he o de
o a ew MeV/c2. This is su icien ly small o ega d he
qua ks as e ec i ely massless. Based on simple geome ical
a gumen s, he magne ic ield di ec ion is always no mal o
he eac ion plane, and he e o e s aigh o wa dly accessible
o expe imen . The chi al sepa a ion e ec (CSE) [21]isa
simila e ec in which he p esence o a ec o cha ge, e.g.,
2469-9985/2016/93(4)/044903(14) 044903-1 ©2016 CERN, o he ALICE Collabo a ion
J. ADAM e al. PHYSICAL REVIEW C 93, 044903 (2016)
elec ic cha ge, causes a sepa a ion o chi ali ies. Fo example,
he p esence o a ne posi i e elec ic cha ge will induce a
posi i e axial cu en along he di ec ion o he magne ic ield,
i.e., igh (le ) handed qua ks mo ing pa allel (an ipa allel) o
he magne ic ield.
The CME can be summa ized in a ela i ely simple
equa ion:

JV=Nce
2π2μA
B, (1)
whe e 
JVis he ec o cu en (elec ic cha ge cu en in his
case), Ncis he numbe o colo s [3 in QCD bu o he numbe s
o colo s, i.e., SU(Nc) gauge ields, a e o in e es in heo y],
μAis he axial chemical po en ial (which encodes he anomaly-
induced chi al imbalance), and 
Bis he magne ic ield. The
CSE can simila ly be summa ized as

JA=Nce
2π2μV
B, (2)
whe e 
JAis he axial cu en ( low o axial cha ges, i.e.,
chi ali ies) and μVis he ec o (elec ic) chemical po en ial.
The coupling be ween hese wo phenomena leads o a wa e
p opaga ion o he elec ic cha ge, esul ing in an elec ic
cha ge quad upole momen o he sys em. This is called he
chi al magne ic wa e (CMW) [22–24]. Impo an ly, o a gi en
(ne ) cha ge s a e o he sys em, he quad upole momen always
has he same sign and is he e o e p esen in an a e age o e
e en s wi h he same ec o cha ge s a e, meaning i may lead
o a signal s ong enough o be obse ed di ec ly in expe imen .
As men ioned abo e, he azimu hal dis ibu ion o pa icles
can be w i en as a Fou ie expansion:
dN
dϕ ∝1+
n
2 ncos[n(ϕ−ψn)],(3)
whe e ϕis he azimu hal angle o he pa icle, nis he Fou ie
coe icien , and ψnis he symme y plane, which in p inciple
can be di e en o each ha monic numbe n.
Taking in o accoun he well-known modula ion o pa icle
emission due o ellip ic low pa ame e ized wi h he Fou ie
coe icien 2(see, e.g., [25]), one can w i e he azimu hal
dis ibu ion o cha ged pa icles due o he CMW [24]as
dN±
dϕ =N±[1 +(2 2∓ A) cos (2(ϕ−ψ2))],(4)
whe e he cha ge asymme y A=(N+−N−)/(N++N−)
is de e mined in some kinema ic egion ( o example in
he expe imen al accep ance), and he pa ame e encodes
he s eng h o he elec ic quad upole due o he CMW.
Bjo ken low [26] ela es he pseudo apidi y o a pa icle o
i s longi udinal p oduc ion poin . Due o space-momen um
co ela ions, he cha ge asymme y A, de e mined in he
expe imen al accep ance, co esponds o he local cha ge
asymme y in a ce ain egion o he i eball.
ALICE measu emen s o he cha ge dependen co ela ions
in sea ch o he CME ha e been published in [27]. This
pape p esen s he ALICE esul s on he cha ge dependen
ellip ic low as a unc ion o he e en cha ge asymme y.
As he e en cha ge asymme y As ongly depends on he
expe imen al accep ance and acking e iciency, we also
p esen ela ed esul s on a di e en ial h ee-pa icle co ela o
ha allows much mo e de ailed s udy o he unde lying physics
mechanisms. We also p esen he co esponding measu emen s
o highe ha monics low ha should be la gely insensi-
i e o he CMW bu sensi i e o he possible backg ound
e ec s.
II. ANALYSIS METHODOLOGY
A. nas a unc ion o Aand in eg al h ee-poin co ela o s
In he heo e ical wo k on he CMW [22–24,28,29]as
well as he analysis published by STAR [30]o Au-Au
collisions a √sNN =200 GeV, he obse able has been he
cha ge-dependen low coe icien ±
nas a unc ion o he
cha ge asymme y A. Expe imen ally, he cha ge asymme y
de ined in a speci ied kinema ic egion mus be co ec ed
o de ec o e iciency, as discussed in [30–32]. The e ec
o he co ec ion is o inc ease he slope o posi i e o nega i e
pa icle ±
n s A. This co ec ion o he slope, hough absolu ely
necessa y, has he undesi ed ea u e o in oducing addi ional
sou ces o sys ema ic unce ain y.
In addi ion o measu ing ±
nas a unc ion o A o a speci ied
e en -selec ion c i e ion (e.g., a cen ali y in e al), one can
measu e he co a iance o ±
nand A, i.e.,  ±
nA−A ±
n,
as a unc ion o some e en -le el a iable (e.g., cen ali y) [33].
The ha monic coe icien is, by de ini ion, n=cos[n(ϕ1−
ψn)], whe e ϕ1is he azimu hal angle o a pa icle in he
e en and ψnis he h ha monic symme y plane. This makes
 ±
nA−A ±
na h ee-poin co ela o . The i s poin is
he low pa icle, he second poin is he e en plane (which
is an es ima o o he ue symme y plane), and he hi d
poin is he e en cha ge asymme y. In cases whe e he e en
plane is de e mined wi h a second pa icle, as is he case wi h
he wo-pa icle cumulan me hod, his co ela o can also be
called a h ee-pa icle co ela o .
The s eng h o he co a iance o any wo a iables is
independen o he sample size as long as he co ela ion
is s a is ically signi ican , so no co ec ion o e iciency is
needed o he h ee-poin co ela o . Fo ha eason, he
co ela ion s eng h is iden ical o he ull se o pa icles
in some collec ion o e en s and o some andomly selec ed
subse o pa icles in he same collec ion o e en s [34].
F om Eq. (4), i ollows ha ±
2≈ 2∓ A/2, which in
u n allows one o w i e  2= −
2− +
2≈ A. By measu ing
 2as a unc ion o A, i is possible o ex ac di ec ly
in expe imen . One can also subs i u e hese e ms in o he
h ee-pa icle co ela o , gi ing
 ±
2A−A ±
2≈∓ (A2−A2)/2=∓ σ2
A/2.(5)
Clea ly, ei he o hese app oaches can be gene alized o
a bi a y ha monic n, yielding in p inciple a di e en
o each ha monic. Because o he symme y o he CMW
e ec , he expec a ion is ha , and he e o e he h ee-pa icle
co ela o , is signi ican only o he second ha monic and
s ongly supp essed o highe ha monics. Since Ais e iciency
dependen , i is necessa y o scale down he obse ed wid h
o accoun o he na u al b oadening due o he binomial
sampling. To calcula e he scale ac o , we compa ed he
044903-2
CHARGE-DEPENDENT FLOW AND THE SEARCH FOR THE . . . PHYSICAL REVIEW C 93, 044903 (2016)
wid hs o Ain a Mon e Ca lo simula ion be o e and a e
econs uc ion e ec s, whe e he ull de ec o esponse was
implemen ed wi hin he GEANT3[35] amewo k. Since he
ALICE acking e iciency is independen o cen ali y, his
scale ac o is also independen o cen ali y. This co ec ion
o σ2
Ain oduces an addi ional sou ce o sys ema ic unce ain y,
in essen ially he same way as he co ec ion o he slope o
±
n s A. In he p esen analysis, his unce ain y ep esen s
a oughly 6% no maliza ion unce ain y on he poin s. Bo h
he co ec ion and he associa ed unce ain y a y depending
on de ec o accep ance and e iciency, analysis cu s, e c.
A key ad an age o measu ing he h ee-pa icle co ela o ,
a he han (di ec ly o indi ec ly), is ha i is e iciency
independen and does no equi e any co ec ion. This educes
he o e all sys ema ic unce ain y on he measu emen , which
be e acili a es compa isons ac oss expe imen s as well as
wi h heo y calcula ions.
In his analysis, wo-pa icle cumulan s a e always used
o calcula e n[which igno es co ela ions no ela ed o
aniso opic low (non low), as well as low luc ua ions]. Then,
he in eg al co ela o is
 ±
nA−A n=cos[n(ϕ1−ψn)] A
−cos[n(ϕ1−ψn)]A
≈cos[n(ϕ1−ϕ2)]A
√cos[n(ϕ1−ϕ2)]
−cos[n(ϕ1−ϕ2)]A.(6)
Fo his equa ion he inne a e age ep esen s an a e age o e
all pa icles in a single e en , and he ou e a e age ep esen s
an a e age o e all e en s. The i s pa icle has he selec ed
cha ge and he second pa icle is o bo h cha ges. Fo he
in eg al co ela o epo ed below, he same pa icles a e used
o calcula e nand A. The slopes ex ac ed om he h ee-
pa icle co ela o we e checked agains he slopes ex ac ed
di ec ly om  2 s Aand ound o be pe ec ly consis en
wi h each o he .
B. Di e en ial h ee-poin co ela o s
A key ad an age o he no el h ee-poin co ela o is ha
i pe mi s mo e di e en ial s udies and as such has mo e
disc imina ing powe . The cha ge asymme y in he e en
can be gene alized o he cha ge o a pa icle in he e en ,
which we will call q3. The a e age o all cha ges in he
e en is equal o he cha ge asymme y, i.e., q3e en ≡A.
Unde his gene aliza ion he co ela o [Eq. (6)] becomes
 nq3−q3 n.
We also use he addi ional no a ion o q31 o deno e he
mean o q3e alua ed when selec ing on he cha ge o he i s
pa icle q1. This is impo an because, by cons uc ion, he
co ela o  nq3−q3 ncon ains educible co ela ions,
i.e., co ela ions ha can be exp essed in e ms o lowe o de
co ela ions [33,34]. These educible co ela ions a e emo ed
by he cons uc ion  nq3−q31 n, which is he e o e a
h ee-poin cumulan .
Using hese ela ions, we es ima e he di e en ial co ela o
in he ollowing way:
 ±
nq3−q31 n
=cos[n(ϕ1−ψn)] q3−cos[n(ϕ1−ψn)]q31
≈cos[n(ϕ1−ϕ2)]q3
√cos[n(ϕ1−ϕ2)]−cos[n(ϕ1−ϕ2)]q31.
(7)
The e alua ion o a di e en ial co ela o is a e y impo an
ea u e o his s udy. Ra he han measu ing only e en
quan i ies, one can also measu e he ela ionship be ween he
low a a pa icula kinema ic coo dina e and he cha ge o
he hi d pa icle a ano he pa icula coo dina e. This means
he e ec can be measu ed as a unc ion o he sepa a ion
in pseudo apidi y o pa icles 1 and 3, o example. This
di e en ial na u e allows o a much mo e de ailed s udy o
he o igin o he co ela ion and p o ides s onge expe imen al
cons ain s on he heo e ical modeling o such e ec s.
Th oughou his pape we use he subsc ip no a ion used
abo e o he i s , second, and hi d pa icles. The cha ge,
azimu hal angle, and pseudo apidi y o he i s pa icle a e q1,
ϕ1, and η1, espec i ely. Simila ly, he azimu hal angle o he
second pa icle is ϕ2, and he cha ge and pseudo apidi y o he
hi d pa icle a e q3and η3, espec i ely.
III. EXPERIMENTAL APPARATUS AND DATA ANALYSIS
ALICE [36,37] is a dedica ed hea y-ion expe imen loca ed
a he La ge Had on Collide a CERN. I is composed o a
wide a ay o de ec o subsys ems. Those used in he p esen
analysis a e he V0 de ec o s, he Inne T acking Sys em (ITS),
and he Time P ojec ion Chambe (TPC). The V0 de ec o s
consis o scin illa o a ays and a e used o igge ing and
cen ali y de e mina ion. The e a e wo V0 de ec o s, V0A and
V0C. The V0A is loca ed 340 cm om he nominal in e ac ion
poin and he V0C is ins alled a 90 cm dis ance in he opposi e
di ec ion. The V0A co e s 2.8<η<5.1 in pseudo apidi y
Obse ed A
0.1−0.05−0 0.05 0.1
{2}
2
0.096
0.098
0.1
0.102
0.104 pos
neg
ALICE
= 2.76 TeV
NN
s
30-40% Pb-Pb
c < 5.0 GeV/
T
p0.2 <
< 0.8
η
-0.8 <
FIG. 1. Ha monic coe icien s +
2( ed squa es) and −
2(blue
ci cles) as a unc ion o he obse ed e en cha ge asymme y A
in he 30–40% cen ali y class. S a is ical unce ain ies only.
044903-3
J. ADAM e al. PHYSICAL REVIEW C 93, 044903 (2016)
Obse ed A
0.1−0.05−0 0.05 0.1
{2}
2
Δ
0.004−
0.002−
0
0.002
0.004
ALICE
= 2.76 TeV
NN
s
30-40% Pb-Pb
c < 5.0 GeV/
T
p0.2 <
< 0.8
η
-0.8 <
/nd = 1.30/8
2
χ
0.00007±In e cep = -0.00009
0.003±Slope = 0.027
Co ec ed A
0.1−0.05−0 0.05 0.1
{2}
2
Δ
0.004−
0.002−
0
0.002
0.004
ALICE
= 2.76 TeV
NN
s
30-40% Pb-Pb
c < 5.0 GeV/
T
p0.2 <
< 0.8
η
-0.8 <
/nd = 1.30/8
2
χ
0.00007±In e cep = 0.00013
0.003±Slope = 0.035
FIG. 2.  2= −
2− +
2as a unc ion o he obse ed (le ) and co ec ed ( igh ) e en cha ge asymme y Ain he 30–40% cen ali y class.
S a is ical unce ain ies only.
and he C side spans −3.7<η<−1.7. The ITS is used o
bo h acking and e ex de e mina ion. The ITS is composed
o h ee subsys ems, each ha ing wo cylind ical laye s o
silicon de ec o s. Each o he laye s co e s a leas |η|<0.9
in pseudo apidi y o ma ch he TPC accep ance. The TPC is he
p ima y acking de ec o a mid apidi y. The TPC is a la ge
gas olume de ec o sepa a ed in o wo egions by a cen al
elec ode, posi ioned in a solenoidal magne ic ield o 0.5 T.
The gas olume is con ained in a cylind ical elec ic ield cage
wi h an inne adius o 85 cm and an ou e adius o 2.5 m,
spanning he ull azimu h 0 <ϕ<2π. I ex ends 5.0 m in he
zdi ec ion, p o iding co e age o he ull adial ack leng h
o pseudo apidi y |η|<0.9.
The p esen manusc ip epo s an analysis o Pb-Pb
collisions a √sNN =2.76 TeV, collec ed by ALICE du ing
he 2010 and 2011 yea s o LHC ope a ions. In he ea ly pa
o he 2010 ope a ion, he Pb-Pb minimum bias (MB) igge
was a 2-ou -o -3 coincidence o (a) signals in wo pixel chips
in he ou e laye o he SPD, (b) a signal in he V0A, (c) a
signal in he V0C. In he la e pa o he 2010 ope a ion and
o all o he 2011 ope a ion, he Pb-Pb MB igge equi ed
a coincidence o bo h V0 de ec o s. The da a sample used in
his analysis comp ises app oxima ely 1.7×107MB igge ed
e en s in he 2010 da a se . In he 2011 se , we use a mix o
he cen al, semicen al, and MB igge s. The cen al igge
is an online igge wi h a h eshold on he mul iplici y in he
V0 de ec o s such ha i co esponds o he 10% mos cen al
e en s. The semicen al igge is de ined simila ly such ha
i co esponds o he 50% mos cen al e en s. The cen ali y
is es ima ed using he mean mul iplici y in he V0 de ec o s,
and he cen ali y is equi ed o be wi hin 5% (absolu e) o
he cen ali y es ima e using he TPC mul iplici y o a oid
mul iplici y luc ua ions in he cen al egion. The longi udinal
posi ion o he p ima y e ex is equi ed o be wi hin 10 cm o
he nominal cen e o he ALICE coo dina e sys em in o de
o ensu e uni o m de ec o accep ance.
T acks a e selec ed in he kinema ic egion |η|<0.8 and
0.2<p
T<5.0GeV/c. They a e equi ed o ha e a leas
70 TPC clus e s, and he pe cen age o hi s o c ossed pad
ows is equi ed o be a leas 80%. The ack i is equi ed
o ha e χ2pe clus e (2 deg ees o eedom) less han 4.0.
Addi ional acking in o ma ion om he ITS is used when i
is a ailable, i.e., when he ack ajec o y in he TPC poin s
o an ac i e a ea o he ITS. The dis ance o closes app oach
o he econs uc ed e ex is equi ed o be wi hin 3.2 cm
in he zdi ec ion and wi hin 2.4 cm in he xy-plane. Due
o he excellen azimu hal uni o mi y o he TPC esponse,
no co ec ion o azimu hal accep ance is needed, no is one
applied. The esul s a e co ec ed o he pTdependence o he
acking e iciency, which is abou 80% a low pTand abou
90% a high pT. The co ec ion p ocedu e is o andomly
exclude acks in such a way ha he e ec i e e iciency is
made o be uni o m in pT. The esul is a 2–3% educ ion in n.
To assess sys ema ic unce ain ies, he analysis is epea ed
o di e en ope a ional condi ions (i.e., he wo o ien a ions
o he expe imen al magne ic ield), di e en e en selec ion
c i e ia, di e en ack selec ion cu s, and di e en ack
econs uc ion me hods. The unce ain ies associa ed wi h
each speci ic selec ion o condi ion a e obse ed o be unco -
ela ed and hus added in quad a u e o de e mine he o e all
Cen ali y (%)
0 10203040506070
〉{2}
2
〈〉A〈 - 〉A{2}
2
〈
0.2−
0.15−
0.1−
0.05−
0
0.05
0.1
0.15
0.2
3−
10×
pos
neg
ALICE
= 2.76 TeV
NN
sPb-Pb
c < 5.0 GeV/
T
p0.2 <
< 0.8
η
-0.8 <
FIG. 3. Th ee-pa icle co ela o o posi i e ( ed squa es) and
nega i e (blue ci cles) pa icles o he second ha monic as a unc ion
o cen ali y. S a is ical (sys ema ic) unce ain ies a e indica ed by
e ical ba s (shaded boxes).
044903-4

CHARGE-DEPENDENT FLOW AND THE SEARCH FOR THE . . . PHYSICAL REVIEW C 93, 044903 (2016)
Cen ali y (%)
0 10203040506070
)〉{2}
2
〈〉A〈 - 〉A{2}
2
〈(
η
/d
ch
dN
0.01−
0.005−
0
0.005
0.01
0.015
0.02
0.025
0.03
neg-pos
ALICE
= 2.76 TeV
NN
sPb-Pb
c < 5.0 GeV/
T
p0.2 <
< 0.8
η
-0.8 <
FIG. 4. Cha ge di e ence o he h ee-pa icle co ela o o
he second ha monic as a unc ion o cen ali y, mul iplied by
dNch/dη[39]. S a is ical (sys ema ic) unce ain ies a e indica ed
by e ical ba s (shaded boxes).
sys ema ic unce ain y. All a o emen ioned sou ces we e
ound o gene ally con ibu e wi h simila magni ude. Many
obse ables epo ed in his manusc ip ha e alues e y close
o ze o, so ha epo ing sys ema ic unce ain ies as a pe cen -
age o hese alues obscu es hei ue meaning. Fo nume ical
s abili y, sys ema ic unce ain ies a e e alua ed as a pe cen age
o  nAo  nq3. This quan i y alone is no necessa ily
physically meaning ul, because i con ains de ec o speci ic
e ec s. I does, howe e , se a na u al scale o he unce ain-
ies. Once unce ain ies a e assessed, hei absolu e alue is
de e mined and hen plo ed oge he wi h he da a poin s.
IV. RESULTS
A. 2 s A
Figu e 1shows +
2and −
2as a unc ion o he obse ed
(unco ec ed) e en cha ge asymme y Ain he 30–40%
cen ali y class. Clea ly isible is an inc ease in −
2wi h
inc easing A, along wi h a co esponding dec ease in +
2.
This is quali a i ely consis en wi h expec a ions om he
CMW [22–24,28,29] as well as wi h he STAR esul s [30].
Visually he ela ionship does no appea exac ly linea o
ei he cha ge. Fo a gi en cen ali y selec ion, he e is some
ange o mul iplici ies. Since Ais a combina ion o numbe s
o pa icles, i is a combina ion o di e en nega i e binomial
dis ibu ions, which a e b oade o lowe numbe s o pa icles.
The e o e, o la ge alues o |A|, one is sampling e en s wi h
lowe mul iplici ies, which can a ec he alue o 2.
Figu e 2shows  2= −
2− +
2as a unc ion o he
obse ed Ain he le panel and o Aco ec ed o e iciency
in he igh panel. To ob ain he co ec ed A, we analyzed
HIJING [38] simula ions p opaga ed h ough a de ec o desc ip-
ion implemen ed in he GEANT3[35] amewo k o de e mine
he ue (gene a ed pa icle le el) Aas a unc ion o he
obse ed ( econs uc ed ack le el) A. I can be seen ha
he e ec o he co ec ion is a modes inc ease in he slope.
Again, hese esul s a e quali a i ely consis en wi h CMW
expec a ions and wi h he STAR da a [30].
B. In eg al co ela o esul s as a unc ion o cen ali y
Conside ing he obse ed inc ease (dec ease) o −
2( +
2)
wi h inc easing A, discussed in he p eceding sec ion, we
expec a posi i e (nega i e) co a iance o −
2( +
2) wi h A, and
indeed his is exac ly wha is seen in he in eg al co ela o .
Addi ionally, i enables a con enien s udy o he e olu ion o
he co ela ion as a unc ion o e en -le el obse ables.
Figu e 3shows he in eg al co ela o o he second
ha monic as a unc ion o cen ali y. A subs an ial inc ease
in he co ela ion s eng h is seen as he collisions become
mo e pe iphe al. This can be caused by a combina ion o
se e al ac o s. The magne ic ield s eng h inc eases as he
impac pa ame e inc eases since he e a e mo e spec a o s
and hus he cu en ge s s onge . This would cause he
co ela ions due o he CMW o ge s onge . Addi ionally,
local cha ge conse a ion (LCC) e ec s could play a ole [33].
I is impo an o no e ha nei he o hese necessa ily comes
a he expense o he o he ; in p inciple he obse able
Cen ali y (%)
0102030405060
〉{2}
3
〈〉A〈 - 〉A{2}
3
〈
0.1−
0.08−
0.06−
0.04−
0.02−
0
0.02
0.04
0.06
0.08
0.1
3−
10×
pos
neg
ALICE
= 2.76 TeV
NN
sPb-Pb
c < 5.0 GeV/
T
p0.2 <
< 0.8
η
-0.8 <
Cen ali y (%)
0102030405060
)〉{2}
3
〈〉A〈 - 〉A{2}
3
〈(
η
/d
ch
dN
0.004−
0.002−
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
neg-pos
ALICE
= 2.76 TeV
NN
sPb-Pb
c < 5.0 GeV/
T
p0.2 <
< 0.8
η
-0.8 <
FIG. 5. Th ee-pa icle co ela o o he hi d ha monic (le panel) o posi i e ( ed squa es) and nega i e (blue ci cles) pa icles, and he
cha ge di e ence mul iplied by dNch/dη( igh panel). S a is ical (sys ema ic) unce ain ies a e indica ed by e ical ba s (shaded boxes).
044903-5
J. ADAM e al. PHYSICAL REVIEW C 93, 044903 (2016)
Cen ali y (%)
0102030405060
〉{2}
4
〈〉A〈 - 〉A{2}
4
〈
0.1−
0.08−
0.06−
0.04−
0.02−
0
0.02
0.04
0.06
0.08
0.1
3−
10×
pos
neg
ALICE
= 2.76 TeV
NN
sPb-Pb
c < 5.0 GeV/
T
p0.2 <
< 0.8
η
-0.8 <
Cen ali y (%)
0 102030405060
)〉{2}
4
〈〉A〈 - 〉A{2}
4
〈(
η
/d
ch
dN
0.004−
0.002−
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
neg-pos
ALICE
= 2.76 TeV
NN
sPb-Pb
c < 5.0 GeV/
T
p0.2 <
< 0.8
η
-0.8 <
FIG. 6. Th ee-pa icle co ela o o he ou h ha monic (le panel) o posi i e ( ed squa es) and nega i e (blue ci cles) pa icles, and he
cha ge di e ence mul iplied by dNch/dη( igh panel). S a is ical (sys ema ic) unce ain ies a e indica ed by e ical ba s (shaded boxes).
could ha e con ibu ions om bo h o hese and/o addi ional
con ibu ions om as ye unknown sou ces o co ela ion.
Local cha ge conse a ion is he p oduc ion o cha ged pai s
a he same space ime poin . Fo Npa icles, he e a e N/2
co ela ed pai s, and N(N−1) combina o ic pai s, meaning
he co ela ion s eng h is p opo ional o N/[N(N−1)], o
app oxima ely 1/N. Figu e 4shows he di e ence be ween
he cha ges o he second ha monic co ela o mul iplied by
dNch/dη[39], whe e dNch/dηis used as a p oxy o he
o al numbe o pa icles, o examine he ole o he dilu ion
o LCC co ela ions on he co ela o . Conside able cen ali y
dependence emains.
The h ee-pa icle co ela o is s udied using o he ha mon-
ics as well. This p o ides impo an addi ional cons ain s
because P- iola ing e ec s a e expec ed o occu wi h espec
o he eac ion plane, he e o e highe ha monics should ha e
e y li le o no co ela ions. The h ee-pa icle co ela o o
he hi d ha monic is shown in Fig. 5and he ou h ha monic
is shown in Fig. 6. In bo h cases he le panel shows he
co ela o o posi i e and nega i e cha ges sepa a ely, and
he igh panel shows he cha ge di e ence o he co ela o
Cen ali y (%)
0 10203040506070
slope pa ame e
0.01−
0
0.01
0.02
0.03
0.04
0.05
0.06
= 2.76 TeV
NN
sALICE, Pb-Pb a
= 200 GeV
NN
sSTAR, Au-Au a
FIG. 7. Slope pa ame e as a unc ion o cen ali y, including
poin s om STAR [30]. S a is ical (sys ema ic) unce ain ies a e
indica ed by e ical ba s (shaded boxes). No shown is he 6%
sys ema ic no maliza ion unce ain y due he MC co ec ion o σ2
A.
mul iplied by dNch/dη. In bo h o hese cases, he cen ali y
dependence o he cha ge dependence is la , in con as o he
second ha monic. This may sugges a di e en na u e o he
co ela ion. I could also e lec a weake cen ali y dependence
o 3compa ed o ha o ellip ic low.
C. Slopes o  2 s A
Figu e 7shows a compa ison be ween slope pa ame e s
es ima ed in his analysis and om he STAR analysis [30]
o Au-Au collisions a √sNN =200 GeV. Fo he STAR
da a, he 2is e alua ed o cha ged pions wi h 0.15 <
pT<0.5GeV/c, in con as wi h he p esen esul s which
a e o uniden i ied had ons wi h 0.2<p
T<5.0GeV/c.
O e all, he slopes a e su p isingly simila when conside ing
he di e en collision ene gies and mul iplici ies, as well as he
di e en kinema ic accep ance (in addi ion o he di e en pT
selec ion, he STAR esul s co espond o he pseudo apidi y
ange |η|<1.0). The STAR da a exhibi a somewha s onge
cen ali y dependence han he ALICE da a. Mo eo e , he
ηΔ
1.5−1−0.5−0 0.5 1 1.5
〉
3
q〈 -
1
〉
3
q〈
2−
1.5−
1−
0.5−
0
0.5
1
1.5
2
3−
10×
= pos ( e lec ed)
1
q
= neg ( e lec ed)
1
q
= pos
1
q
= neg
1
q
20-60% Pb-Pb
= 2.76 TeV
NN
s
ALICE
c < 5.0 GeV/
T
p0.2 <
FIG. 8. Co ela ion be ween he cha ge o he i s pa icle q1and
he cha ge o he hi d pa icle q3. S a is ical unce ain ies only.
044903-6
CHARGE-DEPENDENT FLOW AND THE SEARCH FOR THE . . . PHYSICAL REVIEW C 93, 044903 (2016)
ηΔ
1.5−1−0.5−0 0.5 1 1.5
〉{2}
2
〈
1
〉
3
q〈 - 〉
3
q{2}
2
〈
0.1−
0.08−
0.06−
0.04−
0.02−
0
0.02
0.04
0.06
0.08
0.1
3−
10×
= pos ( e lec ed)
1
q
= neg ( e lec ed)
1
q
= pos
1
q
= neg
1
q
20-60% Pb-Pb
= 2.76 TeV
NN
s
ALICE
c < 5.0 GeV/
T
p0.2 <
FIG. 9. Th ee-pa icle co ela o o he second ha monic, o
posi i e ( ed squa es) and nega i e (blue ci cles) pa icles. S a is ical
(sys ema ic) unce ain ies a e indica ed by e ical ba s (shaded
boxes).
STAR da a exhibi a s onge cen ali y dependence han
p edic ed by he heo e ical models in oking he CMW o
Au-Au a 200 GeV [24]. Addi ionally, hyd odynamical models
ha e been de eloped o a emp o explain he STAR esul s
wi hou in oking he CMW [40,41]. Howe e , no heo e ical
modeling o calcula ions a all, ega dless o mechanism, a e
a ailable o Pb-Pb collisions a √sNN =2.76 TeV.
D. Di e en ial co ela o esul s as a unc ion o η
As discussed abo e, he de ini ion o he h ee pa icle
di e en ial co ela o includes q31, he mean cha ge o he
hi d pa icle when e alua ed wi h a selec ion on q1.The
quan i y q31−q3is shown as a unc ion o η =η1−η3
in Fig. 8. The measu emen s a e pe o med as a unc ion
o |η|and shown as a unc ion o η wi h he poin s
e lec ed abou η =0. This condi ional mean o q3depends
signi ican ly on η and has he opposi e sign when q1is
lipped. The e ec is mos p onounced o η ≈0 and weakes
when η is la ge. When he i s pa icle is nega i e, he
hi d pa icle has a sligh ly posi i e mean cha ge, and when
he i s pa icle is posi i e, he hi d pa icle has a sligh ly
nega i e mean cha ge. No e ha he quan i y q31−q3is
p opo ional o he cha ge balance unc ion [33] and as such
e lec s he cha ge co ela ion leng h.
Figu e 9shows he h ee-pa icle co ela o o he second
ha monic as a unc ion o η. The co ela o exhibi s a a he
non i ial dependence on η: a peak wi h a “ ypical had onic
wid h” o abou 0.5–1 uni s o apidi y and a possible change
o he sign a abou η ≈1 (no e, howe e , hese poin s a e
consis en wi h ze o wi hin he sys ema ic unce ain ies). Bo h
o hose ea u es quali a i ely ag ee wi h possible backg ound
con ibu ion om local cha ge conse a ion combined wi h
s ong adial and ellip ic low [33]. Un o una ely he e exis
no p edic ions o his obse able om he CMW.
The h ee-pa icle co ela o o he hi d and ou h
ha monics as a unc ion o η is shown in Fig. 10. The s eng h
o he co ela ions is signi ican ly educed, by a ac o abou 3
in he case o he hi d ha monic and a leas a ac o o 5 o he
ou h ha monic. The ou h ha monic co ela o is consis en
wi h ze o wi hin e o s. Neglec ing low luc ua ions, he
CMW expec a ions o highe ha monics co ela o s would
be ze o; un o una ely he e a e no eliable calcula ions o he
e ec o low luc ua ions. The (backg ound) con ibu ion due
o he local cha ge conse a ion should oughly scale wi h he
magni ude o he low [33] and is quali a i ely consis en wi h
he expe imen al esul s. Mo e de ailed calcula ions in bo h
scena ios, as well as mo e p ecise da a, a e ob iously needed
o a mo e de ini i e conclusion.
V. SUMMARY AND OUTLOOK
No el h ee-pa icle co ela o s ha e been employed in
an expe imen al sea ch o he CMW. Resul s ha e been
shown o he second, hi d, and ou h ha monic o he
in eg a ed co ela o o he cha ge-dependen low as a unc ion
o cen ali y and he di e en ial co ela o as a unc ion o
pseudo apidi y sepa a ion. A clea dependence o he posi i e
ηΔ
1.5−1−0.5−0 0.5 1 1.5
〉{2}
3
〈
1
〉
3
q〈 - 〉
3
q{2}
3
〈
0.1−
0.08−
0.06−
0.04−
0.02−
0
0.02
0.04
0.06
0.08
0.1
3−
10×
= pos ( e lec ed)
1
q
= neg ( e lec ed)
1
q
= pos
1
q
= neg
1
q
20-60% Pb-Pb
= 2.76 TeV
NN
s
ALICE
c < 5.0 GeV/
T
p0.2 <
ηΔ
1.5−1−0.5−0 0.5 1 1.5
〉{2}
4
〈
1
〉
3
q〈 - 〉
3
q{2}
4
〈
0.1−
0.08−
0.06−
0.04−
0.02−
0
0.02
0.04
0.06
0.08
0.1
3−
10×
= pos ( e lec ed)
1
q
= neg ( e lec ed)
1
q
= pos
1
q
= neg
1
q
20-60% Pb-Pb
= 2.76 TeV
NN
s
ALICE
c < 5.0 GeV/
T
p0.2 <
FIG. 10. Th ee-pa icle co ela o o he hi d (le panel) and ou h ( igh panel) ha monics, o posi i e ( ed squa es) and nega i e (blue
ci cles) pa icles. S a is ical (sys ema ic) unce ain ies a e indica ed by e ical ba s (shaded boxes).
044903-7
J. ADAM e al. PHYSICAL REVIEW C 93, 044903 (2016)
and nega i e pa icle aniso opic low on he e en cha ge
asymme y is p esen ed o di e en cen ali ies in Pb-Pb
collisions. The slopes o his dependence, de e mined by wo
di e en me hods, a e consis en and quali a i ely ag ee wi h
he expec a ions o CMW, as well as simila o hose measu ed
by he STAR Collabo a ion a he op Rela i is ic Hea y
Ion Collide (RHIC) ene gy. The obse ed nonze o signal
in highe ha monics co ela ions indica es a possible s ong
backg ound con ibu ion, likely om LCC in combina ion
wi h s ong adial and aniso opic low. We also ha e p esen ed
esul s on he di e en ial co ela o , which is mo e sensi i e o
he de ail o he unde lying physics and helps o disc imina e
be ween he CMW scena io and he backg ound e ec s. The
second ha monic esul s show a ai ly la ge co ela ion, and he
s eng h o he co ela ion s ongly dec eases wi h inc easing
ha monic numbe . Fu he inpu om heo y is needed o gi e
de ailed cons ain s on he magni ude and ange o backg ound
s CMW co ela ions.
LHC Run 2 will include Pb-Pb collisions a √sNN =
5.02 TeV and will o e subs an ially highe in eg a ed
luminosi y, which will la gely imp o e s a is ical p ecision
o hese measu emen s, and may also help educe some o
he sys ema ic unce ain ies. One o he chie bene i s o
inc eased s a is ical p ecision would be he possibili y o
e alua e he h ee-pa icle co ela o wi h iden i ied pa icles.
The species o bo h pa icle 1 and 3 is o po en ial in e es . The
di e en collision ene gy a ec s bo h he peak s eng h (which
inc eases) and he li e ime (which dec eases) o he magne ic
ield induced in he collision, which p o ides addi ional
in o ma ion. Fo ha eason, an analysis o his co ela o a
lowe collisions ene gies, o example a RHIC, would p o ide
impo an addi ional insigh s.
ACKNOWLEDGMENTS
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 accel-
e 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 ollowing
unding agencies o hei suppo in building and unning
he ALICE de ec o : S a e Commi ee o Science, Wo ld
Fede a ion o Scien is s (WFS), and Swiss Fonds Kidagan,
A menia; Conselho Nacional de Desen ol imen o Cien ´
ı ico
e Tecnol´
ogico (CNPq), Financiado a de Es udos e P oje os
(FINEP), Fundac¸˜
ao de Ampa o `
a Pesquisa do Es ado de
S˜
ao Paulo (FAPESP); Na ional Na u al Science Founda-
ion o China (NSFC), he Chinese Minis y o Educa ion
(CMOE), and he Minis y o Science and Technology o
China (MSTC); Minis y o Educa ion and You h o he
Czech Republic; Danish Na u al Science Resea ch Council,
he Ca lsbe g Founda ion, and he Danish Na ional Re-
sea ch Founda ion; The Eu opean Resea ch Council unde
he Eu opean Communi y’s Se en h F amewo k P og amme;
Helsinki Ins i u e o Physics and he Academy o Finland;
F ench CNRS-IN2P3, he “Region Pays de Loi e”, “Region
Alsace”, “Region Au e gne”, and CEA, F ance; Ge man
Bundesminis e ium u Bildung, Wissenscha , Fo schung
und Technologie (BMBF) and he Helmhol z Associa ion;
Gene al Sec e a ia o Resea ch and Technology, Minis y o
De elopmen , G eece; Na ional Resea ch, De elopmen and
Inno a ion O ice (NKFIH), Hunga y; Depa men o A omic
Ene gy and Depa men o Science and Technology o he
Go e nmen o India; Is i u o Nazionale di Fisica Nuclea e
(INFN) and Cen o Fe mi - Museo S o ico della Fisica e Cen o
S udi e Rice che “En ico Fe mi”, I aly; Japan Socie y o he
P omo ion o Science (JSPS) KAKENHI and MEXT, Japan;
Join Ins i u e o Nuclea Resea ch, Dubna; Na ional Resea ch
Founda ion o Ko ea (NRF); Consejo Nacional de Cienca
y Tecnologia (CONACYT), Di ecci´
on Gene al de Asun os
del Pe sonal Acad´
emico (DGAPA), M´
exico, Ame ique La ine
Fo ma ion Academique - Eu opean Commission (ALFA-EC),
and he EPLANET P og am (Eu opean Pa icle Physics La in
Ame ican Ne wo k); S ich ing oo Fundamen eel Onde zoek
de Ma e ie (FOM) and he Nede landse O ganisa ie oo
We enschappelijk Onde zoek (NWO), Ne he lands; Resea ch
Council o No way (NFR); Na ional Science Cen e, Poland;
Minis y o Na ional Educa ion/Ins i u e o A omic Physics
and Na ional Council o Scien i ic Resea ch in Highe Educa-
ion (CNCSI-UEFISCDI), Romania; Minis y o Educa ion
and Science o Russian Fede a ion, Russian Academy o
Sciences, Russian Fede al Agency o A omic Ene gy, Russian
Fede al Agency o Science and Inno a ions, and The Russian
Founda ion o Basic Resea ch; Minis y o Educa ion o
Slo akia; Depa men o Science and Technology, Sou h
A ica; Cen o de In es igaciones Ene ge icas, Medioambi-
en ales y Tecnologicas (CIEMAT), E-In as uc u e sha ed
be ween Eu ope and La in Ame ica (EELA), Minis e io de
Econom´
ıa y Compe i i idad (MINECO) o Spain, Xun a de
Galicia (Conselle ´
ıa de Educaci´
on), Cen o de Aplicaciones
Tecnol ˜
A³gicas y Desa ollo Nuclea (CEADEN), Cubaen-
e g´
ıa, Cuba, and IAEA (In e na ional A omic Ene gy Agency);
Swedish Resea ch Council (VR) and Knu & Alice Wallenbe g
Founda ion (KAW); Uk aine Minis y o Educa ion and
Science; Uni ed Kingdom Science and Technology Facili ies
Council (STFC); The Uni ed S a es Depa men o Ene gy,
he Uni ed S a es Na ional Science Founda ion, he S a e o
Texas, and he S a e o Ohio; Minis y o Science, Educa ion
and Spo s o C oa ia and Uni y h ough Knowledge Fund,
C oa ia; Council o Scien i ic and Indus ial Resea ch (CSIR),
New Delhi, India; Pon i icia Uni e sidad Ca ´
olica del Pe ´
u.
[1] T. Lee and C.-N. Yang, Ques ion o Pa i y Conse a ion in Weak
In e ac ions, Phys. Re . 104,254 (1956).
[2] C. Wu, E. Amble , R. Haywa d, D. Hoppes, and R. Hudson,
Expe imen al Tes o Pa i y Conse a ion in Be a Decay, Phys.
Re . 105,1413 (1957).
[3] R. Ga win, L. Lede man, and M. Wein ich, Obse a-
ions o he Failu e o Conse a ion o Pa i y and
Cha ge Conjuga ion in Meson Decays: The Magne ic
Momen o he F ee Muon, Phys. Re . 105,1415
(1957).
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