thermal bottomonium suppression at rhic and lhc

Available online at www.sciencedirect.com

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03doNuclear Physics A 879 (2012) 2558www.elsevier.com/locate/nuclphysa

Thermal bottomonium suppression at RHIC and LHC

Michael Strickland a,b,, Dennis Bazow a

a Physics Department, Gettysburg College, Gettysburg, PA 17325, United Statesb Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, D-60438, Frankfurt am Main, Germany

Received 6 January 2012; received in revised form 30 January 2012; accepted 6 February 2012Available online 8 February 2012

bstract

In this paper we consider the suppression of bottomonium states in ultrarelativistic heavy ion collisions.e compute the suppression as a function of centrality, rapidity, and transverse momentum for the states(1s), (2s), (3s), b1, and b2. Using this information, we then compute the inclusive (1s) suppres-on as a function of centrality, rapidity, and transverse momentum including feed down effects. Calculationse performed for both RHIC sNN = 200 GeV AuAu collisions and LHC sNN = 2.76 TeV PbPbllisions. From the comparison of our theoretical results with data available from the STAR and CMS Col-borations we are able to constrain the shear viscosity to entropy ratio to be in the range 0.08 < /S < 0.24.ur results are consistent with the creation of a high temperature quarkgluon plasma at both RHIC and

C collision energies.2012 Elsevier B.V. All rights reserved.

ywords: Quarkonium suppression; Bottomonium suppression; Relativistic heavy ion collision; Quarkgluon plasma

Introduction

The goal of ultrarelativistic heavy ion collision experiments at the Relativistic Heavy Ionollider at Brookhaven National Laboratory (RHIC) and the Large Hadron Collider (LHC) atERN is to create a tiny volume of matter ( 1000 fm3) which has been heated to a tempera-re exceeding that necessary to deconfine quarks and gluons. Lattice quantum chromodynamicsattice QCD) measurements of the equation of state of strongly interacting matter [15] showat there is crossover from plasma at temperatures on the order of 175 MeV which correspondsapproximately two trillion degrees Kelvin. For RHIC sNN = 200 GeV AuAu collisions,

Corresponding author at: Physics Department, Gettysburg College, Gettysburg, PA 17325, United States.E-mail address: [email protected] (M. Strickland).

75-9474/$ see front matter 2012 Elsevier B.V. All rights reserved.

i:10.1016/j.nuclphysa.2012.02.003

26 M. Strickland, D. Bazow / Nuclear Physics A 879 (2012) 2558

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Qqure

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bethTshDpeththitial maximum central temperatures of T0 450 MeV were generated and for current LHCsNN = 2.76 TeV collisions one obtains T0 550 MeV [6]. For the upcoming full energy LHCavy ion runs with sNN = 5.5 TeV one expects T0 700800 MeV.At such extremely high temperatures strongly interacting matter makes a phase transition to

deconfined plasma of quarks and gluons and, as a result, one expects the emergence of Debyereening of the interaction between quarks and gluons. This leads to the dissolution of hadronicund states [7]. A particularly interesting subset of hadronic states consists of those whiche comprised of heavy quarks because the spectrum of such states can be found using potential-sed non-relativistic treatments. Based on such potential models there were early predictions [8,that J/ production would be suppressed in heavy ion collisions relative to the correspondingoduction in protonproton collisions scaled by the number of nucleons participating in thellision.As mentioned above, heavy quarkonium has received the most theoretical attention, since

avy quark states are dominated by short distance physics and can be treated using heavy quarkfective theory. Based on such effective theories of QCD, non-relativistic quarkonium states canreliably described. Their binding energies are much smaller than the quark mass mQ QCD= c, b), and their sizes are much larger than 1/mQ. At zero temperature, since the velocitythe quarks in the bound state is small (v c), quarkonium can be understood in terms ofn-relativistic potential models such as the Cornell potential which can be derived directly from

CD using effective field theory [1012]. Using such non-relativistic potential models studies ofarkonium spectral functions and meson current correlators have been performed [1319]. Thesults have been compared to first-principles lattice QCD calculations [2026] which rely one maximum entropy method [27,28]. Additionally, there have been some lattice developmentsing non-relativistic lattice QCD [29].Additionally, in recent years there has been an important theoretical development, namely the

rst-principles calculation of the thermal widths of heavy quarkonium states which emerge fromaginary-valued contributions to the heavy quark potential. The first calculation of the leading-der perturbative imaginary part of the potential due to gluonic Landau damping was performedLaine et al. [30,31]. Since then an additional imaginary-valued contribution to the potential

ming from singlet to octet transitions has also been computed using the effective field theoryproach [32], and lattice calculations have been performed in order to determine the imaginaryrt of the heavy quark potential [33]. These imaginary contributions to the potential are relatedquarkonium decay processes in the plasma. The consequences of such imaginary parts onavy quarkonium spectral functions [34,35], perturbative thermal widths [30,36], quarkoniumppression in a T-matrix approach [3739], and in stochastic real-time dynamics [40] havecently been studied; however, these studies were restricted to the case of an isotropicthermalasma, which is only the case if one assumes ideal hydrodynamical evolution.The calculation of the heavy quark potential has since been extended to the case of a plasma

ith finite momentum-space anisotropy. Both the real [41,42] and imaginary [4345] parts haveen computed in this case. Additionally, the impact of the imaginary part of the potential on theermal widths of the states in both isotropic and anisotropic plasmas was recently studied [46].he consideration of momentum-space anisotropic plasmas is necessary since, for any finiteear viscosity, the quarkgluon plasma possesses local momentum-space anisotropies [4755].epending on the magnitude of the shear viscosity, these momentum-space anisotropies canrsist for a long time and can be quite large, particularly at early times or near the edges ofe plasma. This is true for both strong and weak coupling values of the shear viscosity and

e magnitude of the maximal momentum-space anisotropies increases with increasing shear

M. Strickland, D. Bazow / Nuclear Physics A 879 (2012) 2558 27

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ththor

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2.

foscosity. In fact, the magnitude of these momentum-space anisotropies can become so largeat they call into doubt the reliability of the viscous hydrodynamical treatment which implicitlysumes a nearly isotropic state.This has motivated the development of a new dynamical formalism called anisotropic hy-

odynamics (AHYDRO) which extends traditional viscous hydrodynamical treatments to caseswhich the local momentum-space anisotropy of the plasma can be large [5155]. The re-

lt is a dynamical framework that reduces to 2nd order viscous hydrodynamics for weaklyisotropic plasmas, but can better describe highly anisotropic plasmas. For one-dimensionalnamics which is homogeneous in the transverse directions, the AHYDRO approach providese temporal and spatial rapidity evolution of the typical hard momentum of the plasma partons,hard, and the plasma anisotropy, . In a previous paper one of us [56] computed the thermal sup-ession of the (1s) and b1 states at LHC energies by folding together the AHYDRO temporalolution of Ref. [55] with results obtained in Ref. [46] for the real and imaginary parts of thending energy. In this paper, we extend this study to compute the suppression of (1s), (2s),(3s), b1, and b2 states at both RHIC and LHC energies.The structure of the paper is as follows. In Section 2 we introduce the model potential we will

e in order to compute the real and imaginary parts of the binding energies of the states undernsideration. The potential used herein is an improved version of the one used in Refs. [56]d [46] and includes the effects of running coupling and an improved parameterization of themerical results for the short-range anisotropic potential. In Section 3 we briefly review themerical method used to solve the Schrdinger equation. In Section 4 we review the AHYDROnamical model we use and discuss the qualitative features we expect to emerge based one resulting dynamical evolution. In Section 5 we present the initial conditions we will usehich include Glauber (or participant) scaling and a two-component model in which we use aear combination of participant and binary collision scaling. In Section 6 we describe how wempute the nuclear modification factor RAA from the spatial and proper-time dependence of theal and imaginary parts of the binding energy. In Section 7 we detail how one can include thefect of feed down from excited states to compute the inclusive or full nuclear modificationctor for the (1s). In Section 8 we present our final results as a function of centrality, rapidity,d transverse momentum. Finally, in Section 9 we present our conclusions and outlook forture work.

Setup and model potential

In this section we specify the two potential models we consider in this paper. We considere general case of a quarkgluon plasma which is anisotropic in momentum space. In the limitat the plasma is isotropic, the real part of the potentials used here reduces to the potentialiginally introduced by Karsch, Mehr, and Satz (KMS) [9] with or without an additional entropyntribution [42] and the imaginary part reduces to the result originally obtained by Laine et. [30]. To begin the discussion we first introduce our ansatz for the one-particle distributionnction subject to a momentum-space anisotropy.

1. Momentum-space anisotropic plasma

The phase-space distribution of gluons in the local rest frame is assumed to be given by the

llowing ansatz [41,5760]


w

diw

than

ex

fointoco

sion

to

w

thal

bu

w

ev

2.

po

1

heonf (t,x,p) = fiso(

p2 + (p n)2/phard), (1)

here fiso is an isotropic distribution which in thermal equilibrium is given by a BoseEinsteinstribution, is the momentum-space anisotropy parameter, and phard is a momentum scalehich specifies the typical momentum of the particles in the plasma and can be identified withe temperature in the limit of thermal isotropic ( = 0) equilibrium.1 The two parameters phardd can, in general, depend on proper time and position; however, we do not indicate thisplicitly for compactness of the notation. The ansatz above is the simplest ansatz which allowsr the breaking of symmetry in the pT pL plane while maintaining local azimuthal symmetrythe transverse directions in momentum space. Note that one can use the same distributiondescribe the quarks in the system [5759] and the quark self-energy in this case has beenmputed explicitly [60]. For our purpose, we are primarily interested in the gluon distribution

nce this will enter into the determination of the heavy quark potential; however, in the sectiondynamics we implicitly assume the same distribution for quark degrees of freedom.Such a breaking of symmetry in the pT pL plane arises naturally in a heavy-ion collision duethe rapid longitudinal expansion of the matter along the beamline direction and the parameterquantifies the degree of momentum-space anisotropy,

= 12p2p2z

1, (2)

here pz p n and p p n(p n) denote the particle momenta along and perpendicular toe direction n of anisotropy, respectively. For heavy ion collisions the anisotropy vector, n, liesong the beamline direction which we generally choose to lie along the z-axis.The energymomentum tensor T (t,x,p) = (2)3 d3p/p0 ppf (t,x,p) for the distri-tion function (1) is diagonal in the comoving frame and its components are [54,61]

E(phard, ) = T = 12(

11 + +

arctan

)Eiso(phard)

R()Eiso(phard), (3a)PT (phard, ) = 12

(T xx + T yy)= 3

2

(1 + (2 1)R()

+ 1)Piso(phard)

RT ()Piso(phard), (3b)PL(phard, ) = T =

3

(( + 1)R() 1

+ 1)Piso(phard)

RL()Piso(phard), (3c)here Piso(phard) is the isotropic pressure and Eiso(phard) is the isotropic energy density. Inerything that follows we will use a conformal equation of state for which Eiso = 3Piso.

2. Model potential

In this subsection we first review the derivation of the short range screened heavy-quarktential in the presence of finite momentum-space anisotropy. The full complex potential for

The only place that we will assume thermal equilibrium herein is in the value of the isotropic Debye mass used in theavy quark potential in Section 2.2.6. In principle, one could use another isotropic distribution function, in which case

e would need to recompute the isotropic Debye mass.


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th

2.

thfu

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w

an

ax

faisotropic plasma was first obtained in Refs. [30,31]. The calculation of the real part of thetential at finite anisotropy was first obtained in Ref. [42] and was later extended to includee imaginary part in Refs. [4345]. After this brief review we construct an analytic approxi-ation to the real part of the heavy quark potential which allows us to compute the potentialficiently without having to resort to complicated two-dimensional numerical integration. Ase will show, the resulting analytic approximation for the real part can be cast into the form ofDebye-screened Coulomb potential with a Debye mass which depends on the relative angle ofe line connecting the quark and antiquark to the anisotropy direction.

2.1. Integral expression for the real part of the short range potentialOne can determine the real part of the heavy-quark potential in the non-relativistic limit from

e Fourier transform of the static gluon propagator. In an anisotropic plasma with a distributionnction given by Eq. (1) at leading order in the strong coupling constant one finds [42]

V (r, ) = g2CF

d3p(2)3

eipr00( = 0,p, ) (4)

= g2CF

d3p(2)3

eiprp2 +m2 +m2

(p2 +m2 +m2 )(p2 +m2)m4, (5)

here g is the strong coupling constant and CF = (N2c 1)/(2Nc) is the quadratic Casimir ofe fundamental representation of SU(Nc). The masses in (5) are given by [42]

m2 = m2D

2p2

(p2z arctan

pzp

2p2 + p2

arctanpz

p2 + p2

), (6)

m2 = m2D( + (1 + ) arctan )(p2 + p2)+ pz

(pz

+ p2(1+)

p2+p2arctan

pz

p2+p2

)

2(1 + )(p2 + p2)

,

(7)

m2 = m2D2

(p2

p2 + p2

1 + 2p2zp2

arctan + pzp

2(2p2 + 3p2)(p2 + p2)

32 p2

arctanpz

p2 + p2

),

(8)

m2 = m2Dpzp|p|4(p2 + p2)

32, (9)

ith mD being the isotropic Debye mass

m2D = g2

22

0

dpp2dfisodp

, (10)

d p2 p2 = p2 + p2z . The above expressions apply when n = (0,0,1) points along the z-is; in the general case, pz and p get replaced by p n and p n(p n), respectively. One canctorize the denominator of (5) by introducing ( ( ) )2m2 M2 M4 4 m2 m2 +m2 m4 , (11)


w

Indithpopaor

atofpoco

2.

re

w

po

T

Tinde

2.

te

w

fi

2ith M2 m2 +m2 +m2 [57]. This allows us to write

V (r, ) = g2CF

d3p(2)3

eiprp2 +m2 +m2

(p2 +m2+)(p2 +m2). (12)

general one must evaluate (12) numerically. The integration can be reduced to a two-mensional integral over a polar angle, , and the length of the three-momentum, p. However,ere can be poles in the integration domain due to the fact that m2 can be negative for certainlar angles and momenta [57].2 These poles are first order and can dealt with using a principle-rt prescription, however, evaluating this integral with the necessary precision requires on theder of 0.5 to 1 seconds per point. This presents a fundamental problem if one intends to evalu-e the potential when solving the Schrdinger equation on large spatial lattices with on the order5123 points. We are, therefore, motivated to find an efficient parametrization of the resultingtential based on a finite set of numerical evaluations. In order to do so, it is necessary to firstnsider various asymptotic limits of the potential.

2.2. Asymptotic limits of the real part of the short range potentialWhen = 0 then m = m+ = mD and all other mass scales are zero. As a consequence, we

cover the isotropic Debye-screened Coulomb potential

lim0V (r, ) = Viso(r) = g

2CF

d3p(2)3

eipr

p2 +m2D= g

2CF4r

er , (13)

here r rmD .In the limit r 0 for arbitrary one finds that the potential reduces to the vacuum Coulombtential [42]

limr0V (r, ) = Vvac(r) = g

2CF

d3p(2)3

eipr

p2= g

2CF4r

. (14)

he same potential emerges for extreme anisotropy since all mi 0 as :lim

V (r, ) = Vvac(r). (15)his is due to the fact that at = the phase space density f (p) from Eq. (1) has support onlya two-dimensional plane orthogonal to the direction n of anisotropy. As a consequence, thensity of the medium vanishes in this limit.

2.3. Subleading terms in the small limitHaving discussed the leading terms in the limits show above, we now discuss the subleading

rms in the small limit. In the limit of small one finds that [57]

m2+ = 1 +

6(3 cos 2 1),

m2 = m2 + m2 =

3cos 2, (16)

here m m/mD and is the angle with respect to the anisotropy vector n. As a result, onends thatThis is related to the presence of unstable collective modes in momentum-space anisotropic plasmas.


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Toeq

w

,

sc

2.

an

be

w

us

w

sclim0V (r, ) = g

2CF

d3p(2)3

eipr

p2 + 2 , (17)

here mD[1+ 6 (3 cos 2 1)]. Expanding the integrand to leading order in and evaluatinge resulting integrals one finds [42]

lim0V (r, ) = Viso(r)

[1 F(r, )], (18)

here Viso(r) is the Debye-screened Coulomb potential in an isotropic medium (13), and thenction F(r, ) f0(r)+ f1(r) cos(2) with

f0(r) = 6(1 er )+ r[6 r(r 3)]

12r2= r

6 r

2

48+ , (19)

f1(r) = 6(1 er )+ r[6 + r(r + 3)]

4r2= r

2

16+ . (20)

e can now define a -dependent screening mass in an anisotropic medium as the inverse of thestance scale rmed() over which |rV (r)| drops by a factor of e:

logVvac(rmed)

V (rmed, ; , T ) = 1. (21)

leading order in this leads to rmed = 1 F(rmed, ). An approximate solution to this lastuation gives [42]

lim0

mD

1 3 + cos 2

16, (22)

here = r1med.With this in hand we have an analytic approximation for the potential in the limit of smallnamely, that it is approximately a Debye-screened Coulomb potential with a -dependent

reening mass given by Eq. (22) such that

lim0V (r, ) Viso(r) =

g2CF4r

er . (23)

2.4. Subleading terms in the large limitWe now turn our attention to the limit of large . For general one can show that in an

isotropic plasma with a distribution function given by Eq. (1) the particle number density canfactorized using a simple change of variables

n(phard, ) =

d3p(2)3

f (t,x,p) = niso(phard)1 + , (24)

here niso is the number density that would be obtained using the isotropic distribution functioned in Eq. (1). Since in an isotropic system one can estimate the screening mass via m2D n/T ,e expect that in the large- limit one can will obtain 2 n(phard, )/phard for the anisotropic

reening mass, which leads to 1/4mD in the large- limit. To see how this emerges


an

la

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F

2.

stvipo

w

Tinlidure

prican

m

sh

w

in

3

co

saalytically we return to the defining equation for the potential given in Eq. (5). In the limit ofrge one finds.3

limV (r, ) = Vvac(r)

4g2CFm

2D

d3p(2)3

eipr

p4. (25)

e can compare this to the small screening-mass expansion of the isotropic potential Debye-reened Coulomb potential

lim0Viso(r)

mD = Vvac(r) g2CF2

d3p(2)3

eipr

p4. (26)

rom the comparison we see that the anisotropic case can be obtained if we identify

lim

mD

21/4. (27)

2.5. Model for the real part of the short range potentialWith both the small- and large- limits of the anisotropic screening mass in hand we can con-

ruct a model for the real part and then compare to direct numerical evaluation of the potentiala Eq. (12). We find that the following form works well to reproduce the r dependence of thetential for all .

(

mD

)4= 1 +

(a 2

b(a 1)+ (1 + )1/8(3 + )b

)(1 + c()(1 + )

d

(1 + e2)), (28)

ith a = 16/2, b = 1/2, d = 3/2, e = 1/3, and

c() = 32 cos(2)+ (9 + 43 46 )2 + 64(6 3)

4(

3(

2 1)2 16(6 3)) . (29)

he value of the parameter a in (28) guarantees that the large- form for the anisotropic screen-g mass (27) is reproduced. The expression for c() is determined by requiring that the small-mit (22) is reproduced. The coefficients b, d , and e were fit by hand in order to optimally repro-ce the anisotropic short-range potential obtained by direct numerical integration. In addition toproducing these limits, the form (28) also guarantees that /mD (1+ )1/4 in the infinitelyolate limit of 1. We emphasize that the form (28) is only a parametrization of the numer-al results which is constructed in such a way as to guarantee the necessary asymptotic limitsd to efficiently reproduce the potential obtained via direct numerical evaluation in an efficientanner. With this parametrization of in hand we can construct a model for the real part of theort range potential for all :

[V (r)]= g2CF4r

er, (30)ith given by Eq. (28).In Fig. 1 we compare the model specified by Eq. (30) with results obtained by direct numerical

tegration for {0.1,1,2,10,100,1000} by plotting the ratio of the potential over the vacuum

Note that the second integral below is infrared divergent and needs to be regulated; however, since we will onlympare the coefficients of such integrals, we do not need to specify how it is regulated as long as we regulate it in the

me manner in each case.


Fian

paCo

Fivi

Con

an

agfrthan

neg. 1. Comparison of the real part of the short range potential obtained from the analytic model specified in Eq. (28)d via direct numerical integration of Eq. (12). Panels (a)(f) show the potential for different values of the anisotropyrameter as indicated in the lower left corner of each panel. In each panel the potential has been scaled by the vacuumulomb potential. Note that the vertical scale changes between panels.

g. 2. Comparison of the real part of the short range potential obtained from the analytic model specified in Eq. (28) anda direct numerical integration of Eq. (12) for = 1.

oulomb potential. This is a very sensitive test of whether or not the parametrization is a goode and as can been seen from Fig. 1 works well over a very large range of possible plasmaisotropies. To see what the actual unscaled potential looks like in Fig. 2 we show the potentialain; however, this time, we do not scale by the vacuum Coulomb potential. As can be seen

om this figure, the model specified by Eq. (30) works extremely well allowing us to expresse short-range anisotropic quarkonium potential as a Debye-screened Coulomb potential withanisotropic screening mass . In the following subsection we will discuss the fact that oneeds to model the long-range potential and construct a model for the potential at all scales.


2.

teshor

ththoffrpohe

w

sc

tespinan

(2

w

co

co

thleor

m

0.co

thteInlihe

co

an

T

4

re2.6. Model for the real part of the potential at all scalesIn order to make a realistic phenomenological model for quarkonium states one must consis-

ntly describe both short and long distance scales. Since heavy quark states are dominated byort distance physics at zero temperature they can be treated using heavy quark effective the-y; however, as the temperature increases one expects the size of the states to increase causinge states to become sensitive to the long range part of the potential. At zero temperature, sincee velocity of the quarks in the bound state is small, quarkonium can be understood in termsnon-relativistic potential models such as the Cornell potential which can be derived directly

om QCD using effective field theory [1012]. A finite temperature extension of the Cornelltential might be provided by the KMS model [9] which describes the free energy of a staticavy quarkantiquark pair in an isotropic plasma via [42,62]

F(r,T ) = g2CF4r

emDr + mD

[1 emDr], (31)

here g is the strong coupling constant, is the string tension, and mD is the isotropic Debyereening mass. Eq. (31) is a model for the action of a Wilson loop of size 1/T and r in themporal and spatial directions, respectively (see [62] and references therein). In the interest ofanning the possibilities for the real part of the potential we define potential model A by equat-g the real part of the potential with the free energy given in Eq. (31). However, in the generalisotropic case we must replace the isotropic screening mass by the anisotropic screening mass8) to obtain

[VA] = F = arer +

[1 er], (32)

here we have replaced g2CF/4 by a phenomenological parameter a in the screened coulombntribution which will be adjusted to match lattice data. Here we take a = 0.385 which isnsistent with the short range part of the heavy quark potential measured on the lattice [63]. Fore isotropic Debye mass, mD , we use m2D = (1.4)2 Nc(1 + Nf /6)4sp2hard/3. The isotropicading-order Debye mass is adjusted by a factor of (1.4)2 in order to take into account higher-der corrections which have been measured in lattice simulations [64]. In the isotropic Debyeass we use a three-loop running for s [65] with MS = 344 MeV which gives s(5 GeV) =2034 in accordance with recent high precision lattice measurements of the running couplingnstant [66]. For the scale of the running coupling we use 2T which is consistent with hardermal loop calculations of quarkgluon plasma thermodynamics [67,68]. Finally, for the stringnsion we use a value of = 0.223 GeV2 which is again obtained from fits to lattice data [63].all cases we use Nc = 3 since we are modeling QCD and take the number of contributing

ght quark flavors to be Nf = 2, which is appropriate for the temperature range consideredrein.4As potential model B we will use the internal energy, U , of the states which has an entropy

ntribution added to it. To achieve this we calculate the full entropy S = F/T using (32)d add T times this to the free energy (32), which leads to the internal energy U = F + T S.

his procedure gives model B for the real part of the heavy quark potential

If one uses instead Nf = 3 the isotropic Debye mass increases by 6% which has only a small effect on the final

sults.


w

pr

2.

lath

w

w

Fosm

th

2.

thdim

teis

w

is[VB ] = U = F T FT

(33)

= ar(1 +r)er + 2

[1 er] rer , (34)

ith given by Eq. (28). In potential model B, we use the same parameters and Debye massescription as used in potential model A.

2.7. Model for the imaginary part of the potentialThe imaginary part of the potential [V ] is obtained from a leading order perturbative calcu-

tion which was performed in the small anisotropy limit [44]. The resulting imaginary part ofe potential is

[V ] = sCFT[(r) (1(r, )+2(r, ))], (35)

here r = mDr , s = g2/(4), CF = (N2c 1)/(2Nc), and

(r) = 2

0

dzz

(z2 + 1)2[

1 sin(zr)zr

], (36)

1(r, ) =

0

dzz

(z2 + 1)2(

1 32

[sin2

sin(zr)zr

+ (1 3 cos2 )G(r, z)])

, (37)

2(r, ) =

0

dz

43z

(z2 + 1)3(

1 3[(

23

cos2 )

sin(zr)zr

+ (1 3 cos2 )G(r, z)])

,

(38)ith being the angle from the beam direction and

G(r, z) = rz cos(rz) sin(rz)(rz)3

. (39)

r numerical efficiency three separate analytic expressions for [V ] which are valid in theall, medium, and large distance limits were determined and used in a piecewise fashion in

eir respective radii of convergence.

2.8. Final potential modelsAs mentioned above, here we consider two potential models, A and B, in which we identify

e potential as coming from the free energy or internal energy, respectively. From both modelsscussed above we will additionally subtract a temperature- and spin-independent finite quarkass correction taken from Ref. [69] which improves the description of charm quark states at lowmperatures, but is a small correction for bottom quarks. The final result for potential model A

VA = [VA] + i[V ] 0.8m2Qr

, model A (40)

ith [VA] given by Eq. (32) and [V ] given by Eq. (35). The final result for potential model B


w

re

ofla

3.

m

vith

on

w

Hm

a

toNre

eq

w

Ifef

an

FVB = [VB ] + i[V ] 0.8m2Qr

, model B (41)

ith [VB ] given by Eq. (34) and [V ] given by Eq. (35). We note that both [VA] and [VB ]duce to the Cornell potential at T = 0 and the short range part (r 1/mD and r 1/ )both reduces to the Coulomb potential, V = a/r , at all temperatures, with a constrained by

ttice data [63].

Solving the 3d Schrdinger equation

To solve the resulting Schrdinger equation we use the finite difference time domainethod [70,71] extended to the case of a complex-valued potential [46]. Here we briefly re-ew the technique. To determine the wave functions of bound quarkonium states, we must solvee time-independent Schrdinger equation

H(x) = E(x),H =

2

2mR+ V (x)+m1 +m2, (42)

a three-dimensional lattice in coordinate space with the potential given by V = [V ]+ i[V ]here the real and imaginary parts are specified in either Eqs. (40) and (41), respectively.ere, m1 and m2 are the masses of the two heavy quarks and mR is the reduced mass,R = m1m2/(m1 + m2). The index on the eigenfunctions, , and energies, E , representslist of all relevant quantum numbers, such as n, l, and m for a radial Coulomb potential. Duethe anisotropic screening mass, the wave functions are no longer radially symmetric if = 0.

evertheless we still label the states as 1S (ground state) and 1P (first p-wave excited state),spectively.To obtain the time-independent eigenfunctions we start with the time-dependent Schrdinger

uation

i

t(x, t) = H(x, t), (43)

hich can be solved by expanding in terms of the eigenfunctions, :

(x, t) =

c(x)eiE t . (44)

one is only interested in the lowest energy states (ground state and first few excited states) anficient way to proceed is to transform (43) and (44) to Euclidean time using a Wick rotation, it :

(x, ) = H(x, ), (45)

d

(x, ) =

c(x)eE . (46)or details of the discretizations used etc. we refer the reader to Refs. [70,71].


3.

atfu

Dstimoflafu

of

w

w

nu

thpepa

3.

tofrHsn

Ast[7

thFosythte1. Finding the ground state

By definition, the ground state is the state with the lowest energy eigenvalue, E0. Therefore,late imaginary time the sum over eigenfunctions (46) is dominated by the ground state eigen-nction

lim(x, ) c00(x)e

E0 . (47)ue to this, one can obtain the ground state wavefunction, 0, and energy, E0, by solving Eq. (45)arting from a random three-dimensional wavefunction, initial(x,0), and evolving forward inaginary time. The initial wavefunction should have a nonzero overlap with all eigenfunctionsthe Hamiltonian; however, due to the damping of higher-energy eigenfunctions at sufficiently

te imaginary times we are left with only the ground state, 0(x). Once the ground state wave-nction (or any other wavefunction) is found, we can compute its energy eigenvalue via

E( ) = |H | | =d3xHd3x

. (48)

To obtain the binding energy of a state, E,bind, we subtract the quark masses and the real partthe potential at infinity

E,bind (E m1 m2 |V()| |

), (49)

here

V() lim|r|[V (, r)

], (50)

hich is a purely real quantity. For an isotropic potential V is independent of the quantummbers and equal to either /mD or 2/mD for potential models A and B, respectively. Ine anisotropic case, however, this is no longer true since the operator V() carries angular de-ndence. Its expectation value is, of course, independent of but does depend on the anisotropyrameter .

2. Finding the excited states

The basic method for finding excited states is to first evolve the initially random wavefunctionlarge imaginary times, find the ground state wavefunction, 0, and then project this state out

om the initial wavefunction and re-evolve the partial-differential equation in imaginary time.owever, there are (at least) two more efficient ways to accomplish this. The first is to recordapshots of the 3d wavefunction at a specified interval snapshot during a single evolution in .fter having obtained the ground state wavefunction, one can go back and extract the excitedates by projecting out the ground state wavefunction from the recorded snapshots of (x, )0,71].An alternative way to select different excited states is to impose a symmetry condition on

e initially random wavefunction which cannot be broken by the Hamiltonian evolution [71].r example, one can select the first p-wave excited state of the (anisotropic) potential by anti-mmetrizing the initial wavefunction around either the x, y, or z axes. In the anisotropic caseis trick can be used to separate the different excited state polarizations in the quarkonium sys-

m and to determine their energy eigenvalues with high precision. This high precision allows


on

dethla

3.

lefrqusioflathun

fothfi

ofthbosu

4.

icthca

busiw

su

thm

re

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inne

ine to more accurately determine the splitting between polarization states which are otherwisegenerate in the isotropic Debye-screened Coulomb potential. Whichever method is used, oncee wavefunction of an excited state has been determined, one can again use the general formu-s (48) and (49) to determine the states binding energy.

3. Results for the binding energies of bottomonium states

In Figs. 3 and 4 we show the real and imaginary parts of the (1s), (2s), (3s), b1, andb2 binding energies as a function of the hard momentum scale, phard, for {0,1,20}. Theft panels show results obtained with potential model A (40) and the right panels show resultsom potential model B (41). In each case we show three different values of . For the bottomark mass we used mb = 4.7 GeV. For the (1s), (2s), and (3s) states we used a lattice

ze of N3 = 2563 with a lattice spacing of a = 0.125 GeV1 0.025 fm giving a lattice sizeL = Na 6.3 fm. For the b1 and b2 states we used a lattice size of N3 = 2563 with a

ttice spacing of a = 0.15 GeV1 0.03 fm giving a lattice size of L = Na 7.6 fm. Noteat the fluctuations seen in some of the data points occur at values of phard where the state isbound. These fluctuations are due to poor convergence of the Schrdinger equation algorithmr unbound states. However, such fluctuations do not enter into our final results because, whene states are unbound (have a negative real part of their binding energy), then we use a largexed decay rate for these states. Details of the precise prescription will be provided in Section 6.

Defining the disassociation scale as the value of phard at which the real and imaginary partsthe binding energy become equal, one finds the values listed in Table 1. As can be seen from

e figures and table one finds that the dissociation scale increases with increasing such thatttomonium states persist longer in a momentum-space anisotropic plasma. Binding energy datach as those presented in Figs. 3 and 4 will be used as input to our suppression calculation.

Dynamical model

In order to describe the spacetime evolution of the system we use anisotropic hydrodynam-s (AHYDRO) which extends traditional viscous hydrodynamical treatments to cases in whiche local momentum-space anisotropy of the plasma can be large [54,55]. The result is a dynami-l framework that reduces to 2nd order viscous hydrodynamics for weakly anisotropic plasmas,t can better describe highly anisotropic plasmas. In this paper we ignore the transverse expan-

on of the matter and model the system as a collection of decoupled (1+1)-dimensional systemsith different initial temperatures; however, we allow for the breaking of boost invariance. Forch effectively one-dimensional dynamics which is homogeneous in the transverse directions,e AHYDRO approach provides the temporal and spatial rapidity evolution of the typical hardomentum of the plasma partons, phard, the plasma anisotropy, , and the four-velocity of thest frame via a hyperbolic angle .We briefly state the setup and final results of Ref. [55] for completeness. The starting point for

e dynamical equations is to assume the same ansatz (1) for the momentum-space anisotropicstribution as was used to compute the heavy quark potential in the previous section. In thecal rest frame of the plasma the ansatz has two parameters phard and . In the boost invariantse phard and would be functions only of proper time; however, in the case of broken boostvariance both phard and becomes functions of proper time, , and spatial rapidity, . Thecessary dynamical equations can be obtained by taking moments of the Boltzmann equation

the relaxation time approximation [55]. The breaking of boost invariance requires that, in


Fisc

po

adthfoor

sug. 3. Real and imaginary parts of the (1s), (2s), and (3s) binding energies as a function of the hard momentumale, phard. The left panels show results obtained with potential model A (40) and the right panels show results fromtential model B (41).

dition to phard and , one must also specify the hyperbolic angle of the local rest frame ofe flow. This can be accomplished by introducing two four-vectors, one of which specifies theur velocity of the local rest frame in lab frame, u, and an additional four-vector, v, which isthogonal to u, i.e. uv = 0. This can be accomplished by introducing a hyperbolic angle ch that

u = (cosh(, ),0,0, sinh(, )), (51a)( )

v = sinh(, ),0,0, cosh(, ) , (51b)


FiThB

TaIsm

St

bb

w

m

4.

m

sug. 4. Real and imaginary parts of the b1 and b2 binding energies as a function of the hard momentum scale, phard.e left panels shows results obtained with potential model A (40) and the right panels show results from potential model(41).

ble 1otropic and anisotropic dissociation scales for the (1s), (2s), (3s), b1, and b2. Dissociation values were deter-ined by finding the value of phard when the real and imaginary parts of the states binding energy become equal.

= 0 = 1ate Potential A Potential B Potential A Potential B(1s) 298 MeV 593 MeV 373 MeV 735 MeV(2s) < 192 MeV 228 MeV < 192 MeV 290 MeV(3s) < 192 MeV < 192 MeV < 192 MeV < 192 MeV1 < 192 MeV 265 MeV < 192 MeV 351 MeV2 < 192 MeV < 192 MeV < 192 MeV 213 MeV

here (, ) is the hyperbolic angle associated with the velocity of the local rest frame aseasured in the lab frame [53]. If the system were exactly boost-invariant then we would have(, ) = at all times.

1. Moments of the Boltzmann equation

In order to obtain the necessary dynamical equations for phard and we follow [55] and takeoments of the Boltzmann equation. For non-boost-invariant (1 + 1)-dimensional dynamics it

ffices to take the zeroth and first moments and project the first moment with either u or v.


FiphTh{0

Tra

w

ofw

tiosm

w

ther

T

thw

viorg. 5. Dynamical parameters as a function of spatial rapidity using a strong coupling value of 4/S = 1. Shown areard (left) and (right) with initial conditions (0, ) = 0 and phard(0, = 0) = 540 MeV with 0 = 0.3 fm/c.e initial phard rapidity dependence is given by a Gaussian profile specified in Eq. (62). Profiles at proper times .3,2.1,3.9,5.7} fm/c are shown.

he result is three coupled partial differential equations which give the proper-time and spatial-pidity evolution of phard, , and :

11 +

( 2(1 + )

) 6

phardphard = 2

[1 R3/4()1 + ], (52a)

R()R() + 4

phardphard

+ tanh( )

(R()R() + 4

phardphard

)

= (

1 + 13RL()R()

)(tanh( ) +

), (52b)

tanh( )(RL()RL() + 4

phardphard

)+ 1

(RL()RL() + 4

phardphard

)

= (

3R()RL() + 1

)( + tanh( )

), (52c)

here R() and RL() are defined in Eqs. (3a) and (3c), respectively. Note that in the derivationthe above equations it was assumed that the system consists of a plasma of massless particles

hich results in a conformal equation of state, i.e. Eiso = 3Piso.The relaxation rate appearing in the first equation (52a) is fixed by requiring that the equa-ns reduce to the evolution equations of second order viscous hydrodynamics in the limit ofall . Doing so gives [54]

= 2T ()5

= 2R1/4()phard

5, (53)

here = /S is the ratio of the plasma shear viscosity to entropy density and we have mappede equilibrium temperature to phard and by requiring that the anisotropic and isotropic en-gy densities are the same, i.e. Eaniso(phard, ) = Eiso(T ), which upon using Eq. (3a) gives=R1/4()phard.We note, importantly, that since the relaxation rate is proportional to phard, one expects

at the relaxation to isotropic equilibrium is slower in regions where phard is lower. In addition,e see that the relaxation rate is inversely proportional to which tells us that when the shearscosity is small we expect to see larger plasma momentum-space anisotropies developing. In

der to illustrate the dependence on initial temperature, in Figs. 5 and 6 we show the evolution


FiphTh{0

Fiar

Th{0

ofas

fran

ofCspw

la

5.

en

ca

stg. 6. Dynamical parameters as a function of spatial rapidity using a strong coupling value of 4/S = 1. Shown areard (left) and (right) with initial conditions (0, ) = 0 and phard(0, = 0) = 350 MeV with 0 = 0.3 fm/c.e initial phard rapidity dependence is given by a Gaussian profile specified in Eq. (62). Profiles at proper times .3,2.1,3.9,5.7} fm/c are shown.

g. 7. Dynamical parameters as a function of spatial rapidity using a strong coupling value of 4/S = 10. Showne phard (left) and (right) with initial conditions (0, ) = 0 and phard(0, = 0) = 540 MeV with 0 = 0.3 fm/c.e initial phard rapidity dependence is given by a Gaussian profile specified in Eq. (62). Profiles at proper times .3,2.1,3.9,5.7} fm/c are shown.

phard and in the case of a strong coupling shear viscosity of = 1/4 for two differentsumed initial central temperatures of 540 MeV and 350 MeV, respectively. As can be seenom these two figures, as the initial temperature decreases, one sees larger momentum-spaceisotropy as expected from Eq. (53). In order to illustrate the dependence on the assumed value in Fig. 7 we show the case of = 10/4 with an initial central temperature of 540 MeV.

omparing Figs. 5 and 7 we see that there is a dramatic increase in the developed momentum-ace anisotropy when changing from 1/4 to 10/4 . The result of these two dependencesill be that we will see less suppression of the bottomonium states when phard is low or isrge.

Initial conditions

In this section we specify the type of initial conditions we use. We study both RHIC and LHCergies, therefore in this section we will present the general formulae which can be used in bothses. In the results section we will specify the specific initial temperatures, collision energies,

arting proper times, etc. that we use in each specific case.


5.

WSanu

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Tan

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Fo20co

np

w

w1. Transverse coordinate dependence

In this paper we will consider collisions of symmetric nuclei, each containing A nucleons.e will study both participant and binary collision type initial conditions [72] using a Woods-xon distribution for each nucleis transverse profile [73]. For an individual nucleon we take thecleon density to be

nA(r) = n01 + e(rR)/d , (54)

here n0 = 0.17 fm3 is the central nucleon density, R = (1.12A1/3 0.86A1/3) fm is theclear radius, and d = 0.54 fm is the skin depth. The density is normalized such thatA

d3r nA(r) = A, where A is the total number of nucleons in the nucleus. The nor-

alization condition fixes n0 to the value specified above. From the nucleon density we firstnstruct the thickness function in the standard way by integrating over the longitudinal direc-n, i.e.

TA(x, y) =

dznA

(x2 + y2 + z2

). (55)

ith this in hand we can construct the overlap density between two nuclei whose centers areparated by an impact parameter vector b which we choose to point along the x direction, i.e.= bx. We choose to locate the origin of our coordinate system to lie halfway between the centerthe two nuclei such that the overlap density can be written as

nAB(x, y, b) = TA(x + b/2, y)TB(x b/2, y). (56)he overlap density will be used later as the probability weight for bottomonium productiond our two-component initial condition. Another quantity of interest is the participant densityhich is given by

npart(x, y, b) = TA(x + b/2, y)[

1 (

1 NNTB(x b/2, y)B

)B]

+ TB(x b/2, y)[

1 (

1 NNTA(x + b/2, y)A

)A]. (57)

r LHC collisions at sNN = 2.76 TeV we use NN = 62 mb and for RHIC collisions at sNN =0 GeV we use NN = 42 mb. From the participant density we construct our first possible initialndition for the transverse phard profile at central rapidity by taking the third root of the rescaledart

pparthard,0 = T0

[npart(x, y, b)

npart(0,0,0)

]1/3, (58)

here T0 is the central temperature obtained in a central collision between the two nuclei.As an alternative initial condition for phard one could use the number of binary collisions

hich is defined asncoll(x, y, b) = NNnAB(x, y, b). (59)


Fiw

Cpaby

w

in

Nce

1.in

5.

prra

tew

en

w

w

nu

mg. 8. Comparison of initial transverse phard profile given by npart scaling and ncoll scaling. A value of NN = 62 mbas used and we show the case of a central collision, i.e. b = 0.

omparisons with RHIC data show that it is necessary to add an admixture of ncoll to the partici-nt, or wounded-nucleon, scaling. We will consider such an admixture as our second possibilitydefining

nmix(x, y, b) = 12 (1 )npart(x, y, b)+ ncoll(x, y, b), (60)ith = 0.145 as fit by the PHOBOS Collaboration [74]. This gives a second possibility for theitial condition for phard at central rapidity

pmixhard,0 = T0[nmix(x, y, b)

nmix(0,0,0)

]1/3. (61)

ote that T0 should be adjusted so that both initial conditions give the same particle density atntral rapidity when integrated over the transverse plane. For = 0.145 we find that T mix0 =079T part0 at LHC energies and T

mix0 = 1.065T part0 at RHIC energies. These values will be used

the results section when we discuss the initial condition dependence of our results.

2. Spatial rapidity dependence

In the previous subsection we fixed two possible prescriptions for the transverse temperatureofile. Since we allow for the breaking of boost-invariance, we also need to give the spatial-pidity dependence in order to complete our specification of the full three-dimensional initialmperature profile. For the number density profile in spatial rapidity ( ) we use a Gaussianhich successfully describes experimentally observed pion rapidity spectra from AGS to RHICergies [7579] and extrapolate this result to LHC energies. The parametrization we use is

n() = n0 exp(

2

2 2

), (62)

ith

2 = 0.64 83

c2s(1 c4s )

ln(

sNN/2mp), (63)

here cs is the sound velocity, mp = 0.938 GeV is the proton mass, sNN is the nucleoncleon center-of-mass energy, and n0 is the number density at central rapidity. We have added a

ultiplicative factor of 0.64 to adjust for broadening of the distribution in rapidity as a function


ofspna

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ofatintw

5.

(6deon

dethan

tiochth

6.

co

an

thtimre

biw

wproper time since the fits, e.g. from [79], were to the final state spectra rather than initial stateectra. In this paper we will use an ideal (conformal) equation of state for which cs = 1/

3 in

tural units.

3. Full three-dimensional initial conditions

We can use Eq. (62) to determine the initial phard rapidity dependence by taking the third rootthe number density. Putting this together with the two possibilities for the transverse temper-

ure dependence determined in the Section 5.1 we can now specify the full three-dimensionalitial temperature profile. Depending on whether we use the number of participants (npart) oro component model (nmix) scaling we have two possible initial phard profiles:

pIhard,0 = T0[npart(x, y, b)e

2/(2 2 )

npart(0,0,0)

]1/3, initial condition I, (64)

pIIhard,0 = T0[nmix(x, y, b)e

2/(2 2 )

nmix(0,0,0)

]1/3, initial condition II. (65)

4. Allowing for initial momentum-space anisotropy

If the initial momentum-space anisotropy is assumed to be zero, i.e. 0 = 0, then Eqs. (64) and5) can be used without modification. However, if 0 = 0 one should require that the same initialnsity profile is obtained. Using the fact that n(phard, ) = niso(phard)/1 + p3hard/

1 +

e finds that this requires T0(0) = (1 + )1/6T0,iso.We must note, however, for completeness sake, that one could also have a non-trivial depen-

nce of the initial anisotropy on the transverse direction and spatial rapidity. In fact, one expectsat towards the transverse and longitudinal edges of the plasma that the initial momentum-spaceisotropies should be larger; however, at this point in time there is no first principles calcula-n of the x and dependence of at the earliest times after the collision, so here we willoose the simplest possibility, which is that it is a constant and equal to zero. We will exploree possibility of finite initial momentum-space anisotropy in future works.

Computing the suppression factor

The AHYDRO time evolution gives us phard and as a function of proper time, transverseordinate x, and spatial rapidity . Solution of the Schrdinger equation gives us the reald imaginary parts of the binding energy of a given state as a function of phard and . Puttingis together gives us the real and imaginary parts of the binding energy as a function of proper

e, transverse coordinate x, and spatial rapidity : [Ebind(,x, )] and [Ebind(,x, )],spectively.If the real part of the binding energy is positive, then the state is bound. If the real part of the

nding energy is negative, then the state is unbound. The imaginary part of the binding energyill give us information about the decay rate of the state in question. To see the exact relationshipe can compute the quantum mechanical occupation number as a function of proper time

n() =(,x)(,x)

( )( )= (x)eiE (x)eiE


w

ra

id

Tw

thstthofbestju

le

w

stthfath

an

dea

tobe

FstInsu= (x)(x)e2[E]= n0e2[E] , (66)

here in the last line we have identified n0 = (x)(x). In order to connect this to the decayte, , we note that is defined empirically through n(t) = n0 exp( ) so that we canentify = 2[E]. Finally, from Eq. (49) we have [Ebind] = [E] so that

(,x, ) ={2[Ebind(,x, )] [Ebind(,x, )] > 0,

10 GeV [Ebind(,x, )] 0. (67)

he value of 10 GeV in the second case is chosen to be large in order to quickly suppress stateshich are fully unbound. We have checked the sensitivity of our results to this value and find thatere is very little dependence on this number as long as it is greater than 1 GeV such that theates are suppressed quickly within the plasma lifetime. In addition, we set the width to zero ife imaginary part of the binding energy is less than zero. Negative values of the imaginary partthe binding energy occur only at large values of and are a result of the small- expansioning applied outside of its range of applicability. Since large corresponds to a (nearly) free

reaming plasma, one expects that the widths should return to their vacuum values ( keV)stifying this choice.We can integrate the instantaneous decay rate, , over proper-time to extract the dimension-

ss logarithmic suppression factor

(pT ,x, ) (f form(pT )

) fmax(form(pT ),0)

d (,x, ), (68)

here form(pT ) is the lab-frame formation time of the state in question. The formation time of aate in its local rest frame can be estimated by the inverse of its vacuum binding energy [80]. Ine lab frame the formation time depends on the transverse momentum of the state via the gammactor form(pT ) = 0form = ET 0form/M where M is the mass of the relevant state and 0form ise formation time of the state in its local rest frame. For the formation times for the (1s),(2s), (3s), b1 and b2 states we take 0form = 0.2, 0.4, 0.6,0.4, and 0.6 fm/c, respectively.We take the initial proper time 0 for plasma evolution to be 0 = 0.3 fm/c at both RHIC

d LHC energies. The final time, f , is defined to be the proper time when the local energynsity becomes less than that of an Nc = 3 and Nf = 2 ideal gas of quark and gluons withtemperature of T = 192 MeV. At this energy density, plasma screening effects are assumeddecrease rapidly due to the transition to the hadronic phase and the widths of the states willcome approximately equal to their vacuum widths.From obtained via Eq. (68) we can directly compute the suppression factor RAA

RAA(pT ,x, ) = e(pT ,x,). (69)or averaging over transverse momenta and implementing any cuts necessary we assume that allates have a 1/E4T spectrum which is consistent with the high-pT spectra measured by CDF [81].tegrating over transverse momentum given pT -cuts pT,min and pT,max we obtain the pT -cutppression factor

RAA(x, ) pT,maxpT,min

dp2T RAA(pT ,x, )/(p2T +M2)2 p . (70)T ,maxpT,min

dp2T /(p2T +M2)2


Foce

w

ov

ov

7.

deco

thp

ef

w

in

8.

thfuR

prthTable 2Feed down fractions extracted from experiment [82] including errors (middle column) and the valuechosen for use herein (right column). Values of fi are constrained such that

i fi = 1.

(1s) productionMechanism % Stat Sys [82] fi used hereinDirect production 50.9 8.2 9.0 0.51 (2s) decay 10.7 7.7 4.8 10.7 (3s) decay 0.8 0.6 0.4 0.8b1 decay 27.1 6.9 4.4 27b2 decay 10.5 4.4 1.4 10.5

r implementing cuts in centrality we compute RAA for finite impact parameter b and mapntrality to impact parameter in the standard manner. For the cuts over centrality and rapidity,e use a flat distribution.In order to compare with experimental observations we should finally average RAA(x, )er x. For this operation we use a production probability distribution which is set by theerlap density specified in Eq. (56)

RAA()

x dx nAA(x)RAA(x, )x dx nAA(x)

. (71)

Excited state feed down

Since a certain fraction of (1s) states produced in high energy collisions come from thecay of excited states, when computing the full (inclusive) RAA for the (1s) one must alsonsider the suppression of the excited states which decay or feed down to it. In order to fixe feed down fractions we use data from

s = 1.8 TeV pp collisions at CDF [82] with a cut

T > 8.0 GeV/c. The resulting feed down fractions are listed in Table 2.

Based on these numbers, we can construct the full (or inclusive) (1s) RAA including thefect of the suppression of excited states via

RfullAA[ (1s)

]= istates

fiRi,AA, (72)

here Ri,AA is the direct suppression of the ith state and the production fractions, fi , are givenTable 2.

Results for RAA

In this section we present our main results which consist of the suppression factors RAA fore (1s), (2s), and (3s), b1 and b2. We will present each states suppression factor as anction of centrality (number of participants) and rapidity. We will then compute the inclusiveAA for the (1s) including the feed effect as described in Section 7. To close the section we willesent the inclusive RAA for the (1s) as a function of transverse momentum and investigate

e sensitivity to the choice of the type of initial conditions used.


FipaBim

8.

Info062co

ina

an

usm

bem

clthg. 9. RHIC suppression factor RAA for the (1s), (2s), (3s), b1, and b2 states as a function of the number ofrticipants (left) and rapidity (right). The top row uses potential model A (40) and the bottom row uses potential model(41). In all plots we used sNN = 200 GeV, assumed a shear viscosity to entropy density ratio of 4/S = 1, andplemented cuts of 0


Filin

ce

duoffrab

do(fw

desu

8.

prsu

wg. 10. RHIC inclusive or full suppression factor RAA for the (1s) including feed down effects. The three differentes correspond to different assumptions for the shear viscosity to entropy ratio 4/S {1,2,3}. In all plots we usedsNN = 200 GeV and implemented cuts of 0


FiSTra

w

imfi

w

peprva

se

re

ex

m

ar

thsm

hiex

w

8.

thfolud

fog. 11. RHIC (1s + 2s + 3s) suppression factor determined via Eq. (74) compared with experimental data from theAR Collaboration [83]. The three different lines correspond to different assumptions for the shear viscosity to entropy

tio 4/S {1,2,3}. In all plots we used sNN = 200 GeV and implemented cuts of 0


Fiofm

an

tras

an

usm

bem

ra

hastprisprth

preng. 12. LHC suppression factor RAA for the (1s), (2s), (3s), b1, and b2 states as a function of the numberparticipants (left) and rapidity (right). The top row uses potential model A (40) and the bottom row uses potential

odel B (41). In all plots we used sNN = 2.76 TeV, assumed a shear viscosity to entropy density ratio of 4/S = 1,d implemented cuts of 0


Fipeth0

to(ien

ca

taprth

8.

inve

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thfititwg. 13. LHC inclusive or full suppression factor RAA for the (1s) including feed down effects compared to ex-rimental data are from the CMS Collaboration [86]. The three different lines correspond to different assumptions fore shear viscosity to entropy ratio 4/S {1,2,3}. In all plots we used sNN = 2.76 TeV and implemented cuts of


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tog. 15. RHIC (left) and LHC (right) inclusive suppression factor RAA for the (1s) including feed down effects compareSTAR [83] and CMS [86] data. In both plots we have fixed 4/S = 2. Collision energies and cuts applied are

dicated in each figure. The solid black line is the result obtained assuming wounded nucleon initial conditions and theshed red line is the result obtained used a two component model with = 0.145. (For interpretation of the referencescolor in this figure legend, the reader is referred to the web version of this article.)

tions build upon a concerted theoretical effort to understand recently obtained RHIC and LHCta on bottomonium suppression [56,8891].We studied two different potential models which were based on the heavy quark free energy) and internal energy (B). We found that the potential based on the free energy gives too muchppression when compared to the available experimental data at both RHIC and LHC energies.n the other hand, results obtained from the potential model that was based on the internal energyem to be in reasonably good agreement with data obtained at both collision energies. We areerefore led to conclude that one should not use potential models based on the free energy.rom the comparison of our theoretical results obtained using the potential based on the internalergy and data available from the STAR and CMS Collaborations we were able to constrain theear viscosity to entropy ratio to be in the range 0.08 < /S < 0.24. We find that our results arensistent with the creation of a high temperature quark-gluon plasma at both RHIC and LHCllision energies.That being said, it is worrisome that one sees such a strong dependence of the results on thetential model used. However, herein we find that at both RHIC and LHC energies a potentialsed on the internal energy seems to better describe the available data with values for the shearscosity to entropy ratio which are consistent with those determined from bulk collective flow.he dependence on the potential used emphasizes the need for a concerted theoretical effort

better determine the heavy quark potential analytically via finite temperature effective fieldeory methods and/or numerically via lattice QCD studies. This will require determination ofe both the real and imaginary parts of the potential at short and long distances and also thependence on the momentum-space anisotropy of the plasma partons. The calculation of theort range part of the potential for arbitrary momentum-space anisotropy is currently underway.In future work we also plan to include the effect of allowing heavy quark states to have a

ow which is decoupled from the soft medium and to include the effect of finite velocities one heavy quark decay rate. This will include the addition of full (3 + 1)d AHYDRO evolution soat we can simultaneously describe elliptic flow and bottomonium suppression. It would also beteresting to investigate the behavior of heavy quarkonium widths near Tc using an AdS/QCDodel. Finally, it will also be necessary to investigate the possibility of pair recombination due

residual spatial correlations among suppressed pairs [92,93]. How these future investigations


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[2[2ill affect the quoted range for /S is a critical open question which will need to be addressed.e leave these interesting questions for future work.

cknowledgements

D. Bazow and M. Strickland were supported by NSF grant No. PHY-1068765. M. Stricklandceived additional support from the Helmholtz International Center for FAIR Landesoffensiver Entwicklung Wissenschaftlich-konomischer Exzellenz program.

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Thermal bottomonium suppression at RHIC and LHC1 Introduction2 Setup and model potential2.1 Momentum-space anisotropic plasma2.2 Model potential2.2.1 Integral expression for the real part of the short range potential2.2.2 Asymptotic limits of the real part of the short range potential2.2.3 Subleading terms in the small limit2.2.4 Subleading terms in the large limit2.2.5 Model for the real part of the short range potential2.2.6 Model for the real part of the potential at all scales2.2.7 Model for the imaginary part of the potential2.2.8 Final potential models

3 Solving the 3d Schrdinger equation3.1 Finding the ground state3.2 Finding the excited states3.3 Results for the binding energies of bottomonium states

4 Dynamical model4.1 Moments of the Boltzmann equation

5 Initial conditions5.1 Transverse coordinate dependence5.2 Spatial rapidity dependence5.3 Full three-dimensional initial conditions5.4 Allowing for initial momentum-space anisotropy

6 Computing the suppression factor7 Excited state feed down8 Results for RAA8.1 Suppression at RHIC energies8.1.1 RAA for (1s+2s+3s) and comparison to STAR data

8.2 Suppression at LHC energies8.3 Transverse momentum dependence8.4 Dependence on the choice of initial condition type

9 Conclusions and outlookAcknowledgementsReferences

thermal bottomonium suppression at rhic and lhc

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