Table Of ContentAzimuthal harmonics of color fields in a high energy nucleus
T. Lappi1,2
1 Department of Physics, P.O. Box 35, 40014 University of Jyv¨askyla¨, Finland
2 Helsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland
Recentexperimentalresultshaverevealedasurprisinglyrichstructureofmultiparticleazimuthal
correlationsinhighenergyproton-nucleuscollisions. Finalstatecollectiveeffectscanberesponsible
formanyoftheobservedeffects,butithasrecentlybeenarguedthatapartofthesecorrelationsare
present already in the wavefunctions of the colliding particles. We evaluate the momentum space
2-particle cumulant azimuthal anisotropy coefficients v {2},n=2,3,4 from fundamental represen-
n
tationWilsonlinedistributionsdescribingthehighenergynucleus. Thesewouldcorrespondtothe
flow coefficients in very forward proton-nucleus scattering. We find significant differences beteen
Wilson lines from the MV model and from JIMWLK evolution. The magnitude and transverse
momentumdependenceofthev {2}valuessuggestthatthefluctuationspresentintheinitialfields
n
5 are a significant contribution to the observed anisotropies.
1
0 PACSnumbers: 24.85.+p,25.75.-q,12.38.Mh,12.38.Lg
2
b
I. INTRODUCTION fluctuations in the nucleus [23] and in the proton [24],
e
F combined with a calculation of the time evolution of the
One of the most surprising results from the LHC initial color fields and eventual matching to a hydrody-
9
proton-nucleus collision experiments have been the kind namical description [25]. We will not attempt to carry
] of azimuthal multiparticle correlation structures [1–7] out this whole program here, but concentrate in this pa-
h (see also RHIC results from deuteron-gold collisions peronlyonapartofit,namelytheanisotropiesproduced
p [8, 9]) that have, in larger collision systems, been at- when a bunch of valence quark-like particles in the fun-
-
p tibuted to hydrodynamical flow. The particle multiplici- damental representation of the gauge group scatter off
e ties in these collisions systems are large enough for some the color field of a large nucleus. The physical picture
h collective effects to take place. Many of these structures (see [20, 21] and more recently [26–29]) in our calcula-
[ have indeed been successfully descibed by hydrodynam- tionisthatofvalencequarksfromtheprobedeflectedin
2 ical calculations [10, 11]. This agreement requires, how- apreferredtransversedirectionbyadomaininthetarget
v ever, a very specific model of the geometry of the initial colorfield. Thisgeneratesamultiparticlecorrelationthat
5 state [12]. It is also not clear whether these small sys- probesthespatialfluctuationsofthetarget. Ourcalcula-
0 temsarewithintheregimeofvalidityofahydrodynami- tion extends the work in [28, 29] in two significant ways.
5 caldescriptionwithrealisticvaluesoftheenergydensity, Firstly, we perform the Fourier-transform from coordi-
5
0 viscosity and system size [13]. nate to momentum space, in order to get an azimuthal
harmoniccoefficientcorrespondingtorealproducedpar-
. The primary collisions leading to energy deposition
1 ticles. Secondly,unlike[28,29],wecorrelatetheparticles
in the central rapidity region are, at the high energies
0
in a given p -bin with a reaction plane determined by
5 reached at the LHC, characterized by very strong non- T
all the produced particles using the 2-particle cumulant
1 linear color fields [14]. These fields are, to leading order
method.
: in the coupling constant, boost invariant. This immedi-
v
atelyleadstothepresenceoflongrangeazimuthalcorre-
i
X lations in particle production [15–22]. In larger collision
r systems, the structure of these correlations in azimuthal II. AZIMUTHAL CORRELATIONS IN CGC
a
angle and transverse momentum is strongly influenced FIELDS
by collective behavior in the later evolution stages of the
system. However, in smaller systems, such as proton-
In the “hybrid formalism” for particle production
nucleus collisions, these collective effects are presumably
in the dilute-dense limit [30–33] the quark spectrum
less significant than in nucleus-nucleus collisions. This
produced in a collision is proportional to the Fourier-
raises the intriguing possibility that in proton-nucleus
transform of the two point function of Wilson lines in
collisionsalsotheazimuthalstructureoftheinitialstage
the color field of the target
colorfluctuationscouldbedirectlyvisibleinthemeasur-
able particle spectrum. dN (cid:90) 1
We do not yet have a very solid quantitative under- d2p ∝ e−pT·(xT−yT)N TrVx†TVyT. (1)
T c
standing of the relative importance of initial color field xT,yT
and later evolution effects for generating anisotropies in
particleproduction. Acompletecalculationofazimuthal The Wilson lines V(x ) in (1) are, in the CGC descrip-
T
anisotropies in this context requires complicated mod- tion, stochastic random SU(3) matrices in the represen-
eling that includes the color field and nucleonic scale tation of the projectile. To calculate the single inclusive
2
cross section one has to average Eq. (1) by the appropri- around an impact parameter b chosen randomly in the
T
ate probability distribution of Wilson lines. transverseplaneofthetarget. Wewillpresentresultsfor
In the leading order CGC treatment that we use differentvaluesoftheparameterBcharacterizingthesize
here, multiparticle correlations can be calculated from of the probe. Note that the coefficients (2) need not be
the higher order moments of the Wilson line operators normalized,sincewewilleventuallydividebytheangular
in Eq. (1). This corresponds to the so called “glasma average spectrum b0 to construct the Fourier harmonic
graphs” [18, 19, 22], which have a very clear interpreta- coefficient. Wewanttocalculatetheangularcorrelations
tion in the hybrid formalism. The target nucleus is rep- withrespecttoaaneventplanedefinedbyalltheallthe
resented by a sheet of color magnetic and color electric produced particles, which form the “reference” that we
fields, which have a domain structure with a character- correlateindividualparticleswith. Thisisdonefollowing
istic length scale 1/Q in the transverse plane. When a the procedure used in the experimental analysis (see e.g.
s
small enough probe (comparable in size to the domain the 2-particle cumulant method in [4]). For this we need
size) hits this target, the resulting particle production is to calculate also the reference coefficients
has a preferred direction given by the direction of the
(cid:90) (cid:90)
color field in the domain. Since this direction fluctuates bn(ref)≡ d2pTeinϕpT e−ipT·(xT−yT)
from event to event, there is of course no anisotropy on
average, but the existence of a preferred direction in in- xT,yT
1
dividual events shows up in a global angular correlation ×S (x −b )S (y −b ) TrV† V (4)
among all of the produced particles, similarly to hydro- p T T p T T Nc xT yT
dynamical flow. We are neglecting here “connected” or
integrated over all momenta.
“BFKL”-like correlations [22], that give rise to a back-
The target Wilson lines are drawn from a completely
to-back peak in the two-particle correlation. These cor-
homogenousandisotropicdistributionthatfillsthewhole
relations are typical “nonflow” correlations that involve
transverselatticewithperiodicboundaryconditions,and
only a few particles, which the experimental analyses of
theprobeisazimuthallysymmetric. Thusthereisnoge-
azimuthal anisotropy try to exclude. We will not discuss
ometrical (i.e. originating in the shape of the probe or
them further here, see however Refs. [22, 34] for more
thetarget)originforazimuthalanisotropypresentinthe
studies on these lines.
calculation. Since the probability distribution of Wilson
It is evident from the above discussion that we expect lines is azimuthally symmetric (although the individual
the correlation to be very sensitive to the transverse size configurations are not), the correlations among the coef-
of the probe. In the case of calculating the initial condi- ficients b are diagonal:
n
tion for an ion-ion collision the probe is large, with the
consequence that the correlation is washed away by the (cid:104)b∗(p )b (q )(cid:105)∝δ , (5)
n T m T m,n
sum over many independent domains in the transverse
where(cid:104)(cid:105)denotesaveragingovertheconfigurationsofWil-
plane. Thus, in contrast to the correlations generated
sonlinesinthetarget. Notethatthesingleparticlespec-
by collective flow, the effect discussed here is stronger in
trum Eq. (1) is explicitly real, configuration by configu-
small collision systems than in large ones.
ration,leadingtob =b∗ . Thiscanbeshownbytaking
The purpose of this paper is to analyze the azimuthal n −n
the complex conjugate of Eq. (1) and exchanging the in-
correlationstructureofparticleproductionusingEq.(1)
tegration variables x and y . The two particle pair
in more detail. In particular, we want to study its de- T T
correlation function is now
pendenceontheharmonicn,transversemomentum,and
thetransversesizeoftheprobe. Thepracticalprocedure ∞
used here is the following. We first divide the pT range ddN∆pϕair ∝ (cid:88) (cid:104)b∗n(pT)bn(qT)(cid:105)cos(n∆ϕ). (6)
accessible on the lattice into bins. We use here 50 bins, n=−∞
but we have checked that the results are independent of
FromthiswecanidentifythecorrelationfunctionFourier
thesizeofthebin. WethendefinetheFouriercoefficient
coefficients (using the notation of [4])
of the single particle spectrum as
(cid:104)b∗(p )b (q )(cid:105)
(cid:90) (cid:90) Vn∆(pT,qT)= (cid:104)bn∗(pT)bn(qT)(cid:105), (7)
bn(pT)≡ d2pTeinϕpT e−ipT·(xT−yT) 0 T 0 T
|pT|∈bin xT,yT and define the 2-particle cumulant azimuthal harmonic
1 as in [4] as
×S (x −b )S (y −b ) TrV† V . (2)
p T T p T T Nc xT yT V (p ,ref)
v {2} = n∆ T (8)
n (cid:112)
V (ref,ref)
The transverse coordinate profile of the probe has been n∆
taken as a Gaussian (cid:104)b∗n(pT)bn(ref)(cid:105)
= (cid:104)b∗0(pT)b0(ref)(cid:105) . (9)
S (x −b )=exp(cid:26)−(xT −bT)2(cid:27) (3) (cid:114)(cid:104)b∗n(ref)bn(ref)(cid:105)
p T T 2B (cid:104)b∗0(ref)b0(ref)(cid:105)
3
A nice interpretation of Eq. (9) can be obtained by writ- thing to do, since the pair correlation function Eq. (6) is
ing it as a product of three terms, the correct inclusive observable to be obtained via the
target average [17, 38].
R (p ,ref)
v {2}=v {bp} n T . (10)
n n R (p ,ref)
0 T
III. RESULTS AND DISCUSSION
Here we denote by
(cid:104)b∗(p )b (p )(cid:105)
v {bp}2 = n T n T (11) The numerical evaluations of the first four anisotropy
n (cid:104)b∗0(pT)b0(pT)(cid:105) coefficientsareshowninFigs.1,2and3. Theresultsare
presented in scaling units, as v plotted against p /Q
the flow coefficient for particles in the p bin with re- n √ T s
T
for different size probes, i.e. different BQ . Also
spect to the event plane of that p bin (“bp” stands for s
T
shown are the “bin plane” coefficients v {bp}, defined
“bin plane”). This is the equivalent (although here in n
by Eq. (11). To set the scale of the parameters in phys-
momentum, not position space) of the quantity calcu-
ical units we note that the fundamental representation
lated in [28]. This is then corrected by two “correlation
saturation scale around midrapidity at the LHC should
coefficients”. The first one is the correlation coefficient
be [39] around Q ∼ 1 GeV and the typical size of a
between the reference reaction plane and the p -bin re- s
T
proton in hard particle production at small x around
action plane:
B ≈ 4 GeV−2 [40]. Thus a realistic probe size for LHC
√
R (p ,ref)≡ (cid:104)b∗n(pT)bn(ref)(cid:105) ≤1, pA collisions would very roughly be BQs ≈2.
n T (cid:112)
(cid:104)b∗(p )b (p )(cid:105)(cid:104)b∗(ref)b (ref)(cid:105) Thefirstimmediateobservationfromthenumericalre-
n T n T n n
(12) sults is that the color field fluctuations indeed generate
where the inequality follows from the Schwartz inequal- anisotropiesthatarelarge,oftheorderoftheexperimen-
ity. The interpretation of this correction is clear: for tallymeasuredanisotropycoefficients. Itseemstherefore
a fixed anisotropy with respect to the p -bin reaction plausible that the color field fluctuations do play a size-
T
plane, a decorrelation of the p -bin reaction plane from able role in the observed anisoptropy in small systems,
T
thereferencereactionplanedecreasestheflowcoefficient and must be taken into account together with the flow
v {2}. The other correlation coefficient factor in (10) contribution. Also the momentum distribution has the
n
samestructureastheobservedtransversemomentumde-
(cid:104)b∗(p )b (ref)(cid:105)
R (p ,ref)≡ 0 T 0 ≤1 pendence of the flow, first rising until ∼ Qs and then
0 T (cid:112)
(cid:104)b∗(p )b (p )(cid:105)(cid:104)b∗(ref)b (ref)(cid:105) decreasing. The “bin plane” coefficients v {bp} do not
0 T 0 T 0 0 n
(13) decrease nearly as fast at high momentum, from which
is related to the multiplicity and appears in the denom- one can deduce that the decrease of the anisotropy co-
inator, increasing v {2}. This can be understood as fol- efficients at large p follows from the decorrelation of
n T
lows: with larger fluctuations in the p -bin multiplicity the event plane in the p bin from the reference. This
T T
thatareindependentofthereferencemultiplicity,afixed explains why this decrease was not seen in [28], where
correlation between b (p ) and b (ref) implies a larger this decorrelation was not taken into account. The MV
n T n
correlationbetweenp -binandreferencereactionplanes. modelhasagluonspectrumthatismoresharplypeaked
T
In other words, since b ∼ v b , for a given correlation around Q , i.e. a narrower distribution of different size
n n 0 s
between(ref andp )b ’s,thesmallerthecorrelationbe- color field domains. This shows up in significantly larger
T n
tween b ’s, the larger must the correlation between v :s values for the v coefficients. The main effect of JIMLK
0 n n
be. evolutionistoaddmoresmallcolorfielddomains(larger
We take the Wilson lines V(x ) appearing in Eq. (1) p gluons), which decrease the anisotropy of the particle
T T
either from the MV model or resulting from JIMWLK spectrum.
evolution of the distribution of Wilson lines. Both are Thereis,however,animportantcaveatconcerningany
discretized on a 10242 transverse lattice. For the MV directcomparisonoftheseresultstoexperimentalvalues.
model we use a (fundamental representation) saturation Namely, we were considering, in Eq. (1), only incoming
scale of Q a = 0.119, where a is the lattice spacing. quarks. For antiquarks one must replace the Wilson line
s
The JIMWLK calculation starts with a MV model at by its Hermitian conjugate, which changes the sign of b
n
Q a = 0.0220 and, after y = 10 units of evolution in ra- for odd n. Away from the very forward valence region
s
pidity(withrunningcoupling)endsupwithQ a=0.117. in the probe, there are an approximately equal amount
s
TheMVmodelWilsonlinesareconstructedfollowingthe of quarks and antiquarks present, with contributions to
proceduredecribedinmoredetailin[35]andtherunning v that therefore cancel. Gluons do not have nonzero
3
coupling JIMWLK evolution performed using the algo- odd harmonics in this mechanism, because the adjoint
rithmof[36]. Theparametervaluesusedhereareexactly representation is real and thus odd b ’s vanish. Any odd
n
the same as for the 10242-lattice in [37]. Note that we harmonic surviving in the final state around midrapidity
are only averaging two-point functions of the coefficients must therefore have an origin that is different from the
b , not ratios ∼b /b . This makes the averaging proce- one discussed here.
n n 0
durenumericallyquitestableandisphysicallythecorrect Theotherwordofcautionininterpretingtheseresults
4
JIMWLK Q √Β = 1.8 MV Q √Β = 1.8
s s
Q √Β = 2.6 Q √Β = 2.6
s s
0.2 Q √Β = 3.9 0.2 Q √Β = 3.9
s s
2 2
v v
1 1
0. 0.
0 0
0 1 2 3 4 5 0 1 2 3 4 5
p / Q p / Q
T s T s
FIG. 1: Second harmonic coefficient v {2} calculated with JIMWLK-evolved (left) and MV-model (right) Wilson line config-
2
urations. Thethinlinesrepresentthecofficientsv {bp}(seeEq.(11))calculatedwithrespecttotheeventplaneinthep bin
2 T
only.
5 JIMWLK Q √Β = 1.8 5 MV Q √Β = 1.8
1 s 1 s
0. Q √Β = 2.6 0. Q √Β = 2.6
s s
Q √Β = 3.9 Q √Β = 3.9
1 s 1 s
0. 0.
3 3
v v
5 5
0 0
0. 0.
0 0
0 1 2 3 4 5 0 1 2 3 4 5
p / Q p / Q
T s T s
FIG. 2: Third harmonic coefficient v {2} calculated with JIMWLK-evolved (left) and MV-model (right) Wilson line configu-
3
rations. The thin lines represent the cofficients v {bp} (see Eq. (11)) calculated with respect to the event plane in the p bin
3 T
only.
5 JIMWLK Q √Β = 1.8 5 MV Q √Β = 1.8
1 s 1 s
0. Q √Β = 2.6 0. Q √Β = 2.6
s s
Q √Β = 3.9 Q √Β = 3.9
1 s 1 s
0. 0.
4 4
v v
5 5
0 0
0. 0.
0 0
0 1 2 3 4 5 0 1 2 3 4 5
p / Q p / Q
T s T s
FIG. 3: Fourth harmonic coefficient v {2} calculated with JIMWLK-evolved (left) and MV-model (right) Wilson line config-
4
urations. Thethinlinesrepresentthecofficientsv {bp}(seeEq.(11))calculatedwithrespecttotheeventplaneinthep bin
4 T
only.
5
is related to the dependence on the size of the probe, gluon fields in a high energy nucleus, as they are seen
√
parametrized here by the width of the Gaussian B. As by a small probe consisting of valence-like quarks. We
anticipated,themagnitudeofthecorrelation,anditsde- also quantified here the effect of correlating the parti-
pendenceonthetransversemomentum,dependsstrongly cleswiththeeventplanedeterminedbyalltheproduced
on the size of the interaction region. Although one can particles, using the two-particle cumulant method at the
quite well estimate this, it depends on nonperturbative partonlevel. Thequantitativeresultsstronglydependon
physicsintheprotonandcannotultimatelybecontrolled the details of the p -distribution of gluons in the CGC
T
in a weak coupling calculation. wavefunction and on the transverse size of the probe.
Results for azimuthal correlations in a full CYM sim- However, all the results show large contributions to the
ulation have also been presented recently by Schenke, harmonicsfromthesepurelyinitialstateeffects. Forodd
Schlichting and Venugopalan [41, 42]. Their calculation harmonics they largely cancel between quarks and anti-
includes effects of both color field and nucleonic fluctu- quarks, but for even harmonics these are sizeable effects
ations in the probe proton and the target nucleus. The that need to be considered when interpreting the exper-
probe and target geometries also have a significant ef- imental results from proton-nucleus collisions.
fect through the CYM pre-equilibrium version of the
usual hydrodynamical mechanism that converts spatial
anisotropy to momentum space, leading to also odd har-
monics. These geometrical effects have not been in-
Acknowledgements
cluded in our work, which should therefore not be com-
pared directly with experimental data. Our focus here
has been, in stead, on quantifying the generic obser- The author wishes to thank A. Dumitru for discus-
vation that fluctuating color fields result in azimuthal sions, K. J. Eskola for comments on the manuscript and
anisotropiesinmultiparticlecorrelations, evenintheab- BaruchCollege,CUNYforhospitalityintheinitialstages
sence of anisotropies in the impact parameter depen- of this work. The author is supported by the Academy
dence. of Finland, projects 267321 and 273464. This work was
As a conclusion, we have here studied the momentum done using computing resources from CSC – IT Center
spaceazimuthalanisotropystructureofthe“colorglass” for Science in Espoo, Finland.
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