Thermal Width of the � at Large ’t Hooft Coupling

Jorge Noronha1 and Adrian Dumitru2,3,4

1Department of Physics, Columbia University, 538 West 120 Street, New York, New York 10027, USA2Department of Natural Sciences, Baruch College, CUNY, 17 Lexington Avenue, New York, New York 10010, USA

3The Graduate School and University Center, City University of New York, 365 Fifth Avenue, New York, New York 10016, USA4RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA

(Received 24 July 2009; published 8 October 2009)

We use the anti–de Sitter/conformal field theory correspondence to show that the heavy quark (static)

potential in a strongly coupled plasma develops an imaginary part at finite temperature. Thus, deeply

bound heavy quarkonia states acquire a small nonzero thermal width when the ’t Hooft coupling � ¼g2Nc � 1 and the number of colors Nc ! 1. In the dual gravity description, this imaginary contribution

comes from thermal fluctuations around the bottom of the classical sagging string in the bulk that connects

the heavy quarks located at the boundary. We predict a strong suppression of �’s in heavy-ion collisions

and discuss how this may be used to estimate the initial temperature.

DOI: 10.1103/PhysRevLett.103.152304 PACS numbers: 11.25.Tq, 12.38.Mh, 24.85.+p

The conjectured equivalence of strongly coupled four-dimensional N ¼ 4 supersymmetric Yang-Mills (SYM)theory to type IIB string theory on AdS5 � S5 [1] has led tonew insight into the strong coupling dynamics of large Nc

gauge theories at finite temperature. In fact, the anti–de Sitter/conformal field theory (AdS/CFT) correspon-dence has been particularly useful to compute real timecorrelators of gauge invariant quantities in stronglycoupled plasmas such as 2-point functions of the energy-momentum tensor at finite temperature [2]. For instance, itwas shown that the shear viscosity to entropy density ratiosatisfies 4��=s � 1 in all strongly coupled gauge theoriesthat possess a dual description in terms of supergravity [3].Here, we adopt strongly coupled N ¼ 4 SYM theory asa toy model for the deconfined, high-temperature phaseof QCD.

The dual description of the gauge theory at finite tem-perature involves a near-extremal black brane in the bulk,which leads to a five-dimensional metric (in real time)given by

ds2 ¼ �G00ðUÞdt2 þGxxðUÞd~x2 þGUUðUÞdU2: (1)

G00ðUhÞ ¼ 0 defines the location Uh of the black branehorizon in the 5th coordinate, and the boundary is atU ! 1.

The potential between fundamental static sources sepa-rated by a distance L at large ’t Hooft coupling � in N ¼4 SYM was computed in [4,5] and shown to be propor-tional to 1=L (due to conformal invariance of the theory)

and toffiffiffiffi�

p, which indicates that charges are partially

screened even in vacuum [4,6]. In general, thermal effectsare expected to reduce the binding energies of small statesof very heavy quarks at high T. At strong coupling thermalscreening corrections appear at the same order in � as thevacuum potential [7], as opposed to Debye screening inweakly coupled quark-gluon plasmas [8]. However, at

distances L < 1=T these corrections are suppressed by afactor of ðLTÞ4, which originates from the behavior of thedual bulk geometry near the black brane horizon. Thermaleffects also diminish as �=s increases [9].In this Letter we show that at finite temperature the static

potential in a strongly coupled plasma develops an imagi-nary part due to fluctuations about the extremal configu-ration, which corresponds to a string connecting the fun-damental sources at the boundary of the geometry. Such animaginary part arises also in perturbative quantum chro-modynamics (pQCD) at order�g4 due to Landau dampingof the static gluon exchanged by the heavy quark sources[10]. At large ’t Hooft coupling, however, it appears al-ready at the same order as the vacuum potential, i.e., at

Oð ffiffiffiffi�

p Þ. Therefore, energy levels in this potential are notsharp because they acquire a thermal width, E ¼ Evac þ�ET � i�. The width � is smaller than the vacuum energyEvac if LT < 1 (which is the relevant regime in the limit ofvery heavy quarks,mQ ! 1) and of the same order in both

� and LT as the shift �ET of the real part of the potentialdue to thermal screening effects. We propose that theimaginary part of the potential mentioned above can beobserved experimentally via the suppression of � to dilep-ton decays in heavy-ion versus pþ p collisions at RHICand LHC.The relevant operator for our discussion is the path-

ordered Wilson loop defined as

WðCÞ ¼ 1

Nc

TrPeiR

A�dx�

; (2)

where C denotes a closed loop in the boundary, A� is thenon-Abelian gauge field, and the trace is over the funda-mental representation of SUðNcÞ. We consider a rectangu-lar loop with one direction along the time coordinate t andspatial extension L. In the asymptotic limit t ! 1, thevacuum expectation value of the loop defines a static

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potential via hWðCÞi � e�itVQ �QðLÞ. This is what we call the‘‘heavy quark potential.’’

The expectation value of WðCÞ can be calculated atstrong ’t Hooft coupling in non-Abelian plasmas at largeNc that admit a weakly coupled dual gravity descriptionaccording to AdS/CFT [1]. More specifically,

hWðCÞiCFT ¼ Zstring; (3)

where Zstring is the full supersymmetric string generating

functional, which is defined in a 10 dimensional back-ground spacetime and includes a sum over all the stringworld sheets whose boundary coincide with C. In thesupergravity approximation � ¼ g2Nc � 1 and Nc ! 1and, in this case, an infinitely massive excitation in thefundamental representation of SUðNcÞ in the CFT is dual toa classical string in the bulk hanging down from a probebrane at infinity [4,5]. Within this approximation Zstring �eiSNG and the dynamics of the string is given by the classicalNambu-Goto (NG) action (we neglect the contributionfrom other background fields such as the dilaton)

SNG ¼ � 1

2��0Z

d2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� dethab

p(4)

where hab ¼ G��@aX�@bX

� (a, b ¼ 1, 2), G�� is the

background bulk metric,�a ¼ ð�; �Þ are the internal worldsheet coordinates, and X� ¼ X�ð�; �Þ is the embedding ofthe string in the 10-dimensional spacetime. For N ¼ 4SYM, the configuration that minimizes the action is aU-shaped curve that connects the string end points at theboundary and has a minimum at some U� in AdS5 [4,5].

The induced metric is

dethab ¼ X02 � _X2 � ð _X � X0Þ2; (5)

where X0�ð�;�Þ ¼ @�X�ð�; �Þ and _X�ð�;�Þ ¼

@�X�ð�;�Þ. We choose a gauge where the coordinates of

the static string are X� ¼ ðt; x; 0; 0; UðxÞÞ, where � ¼ tand � ¼ x. We neglect the string dynamics in the five-dimensional compact space and perform the calculation inreal time at the boundary. In fact, fixing the extremalconfiguration in such a way implies the t ! 1 limit; whilea Wick rotation is of course still possible (by switching toEuclidean metric, which would still give a complex expec-tation value for theWilson loop) one can no longer performan analytic continuation to imaginary time where the ex-pectation value should be real.

In this gauge,

SNG ¼ � T2��0

Z L=2

�L=2dx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU02 þ VðUÞ

q; (6)

where T ! 1 is the total time interval. Note that we haveassumed that G00GUU ¼ 1 (which is in general valid when� ! 1) and defined VðUÞ � G00Gxx, which satisfiesVðUÞ � 0 for U 2 ½Uh;1Þ. The equations of motion ob-tained from Eq. (6) determine the classical string profile

UcðxÞ as discussed in detail in Refs. [4,5,7,9]. The solutionx ¼ xðUcÞ satisfies the following boundary condition:

L

2¼

Z 1

U�dU

�VðUÞ

�VðUÞVðU�Þ � 1

���1=2; (7)

which is used to obtain U� ¼ U�ðL; TÞ. The expectationvalue of the Wilson loop is obtained by substituting theclassical string solution UcðxÞ into the action in Eq. (6). Ingeneral, when LT 1 the dominant contribution to thepotential comes from the extremal world sheet configura-tion described above and the potential is computed as aseries in LT. Other configurations are expected to contrib-ute significantly when LT > 1 [11]. However, L < 1=T isin fact the region of interest for bound states of very heavyquarks which have small radii.The heavy quark potential in the vacuum of N ¼ 4

SYM has the following simple analytical form (after sub-tracting the self-energy contribution from the infinitelymassive quarks) [4]

V Q �QðLÞ ¼ � 4�2

�ð1=4Þ4ffiffiffiffi�

pL

; (8)

which may be compared to the standard SUðNcÞ Coulombpotential at large Nc corresponding to weak coupling:

V CoulðLÞ ¼ � 1

8�

g2Nc

L: (9)

We now take into account thermal fluctuations aroundthe classical solution UcðxÞ. We shall show that the Wilsonloop develops an imaginary part due to fluctuations nearthe bottom of the classical string configuration U�. Weconsider long wavelength fluctuations of the string profileUcðxÞ ! UcðxÞ þ UðxÞ (with U0 ! 0), which give theleading contribution to the string partition function in thesupergravity approximation as follows:

Zstring �Z

DX�eiSNGðXÞ �Z

DUðxÞeiSNGðUcþUÞ: (10)

When the bottom of the classical string is sufficiently closeto the horizon (though still above it), the world sheetfluctuations, UðxÞ, near x ¼ 0 (where U0

c ¼ 0) canchange the overall sign of the argument of the NG squareroot and generate an imaginary contribution to the action.In this case, both U02

c and VðUcÞ are small and the NGsquare root cannot be expanded in powers of U.The integral over UðxÞ is performed by dividing the x

interval in 2N parts such that xN ¼ L=2, x�N ¼ �L=2,xj ¼ j�x:

Zstring�ZdðU�NÞ���dðUNÞe

�iðT�x=2��0ÞPj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU02

j þVðUjÞp

:

(11)

Near x ¼ 0 we expand UcðxjÞ ’ U� þ x2jU00c ð0Þ=2 and

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U02j þ VðUjÞ ’ C1x

2j þ C2, with

C1 ¼ 12U

00c ð0Þð2U00

c ð0Þ þ V0� þ UV 00� þ 12U

2V000� Þ’ 1

2U00c ð0Þð2U00

c ð0Þ þ V 0�Þ � 0

C2 ¼ V� þ UV 0� þ 12U

2V 00� :

(12)

Here, V� ¼ VðU�Þ, V 0� ¼ V 0ðU�Þ and so on. The imaginarypart of the Q �Q potential arises from the region of Uwhere C2 < 0.

We isolate the contribution to the path integral fromx ¼ xj,

Ij �Z Ujmax

Ujmin

dðUjÞ exp��i

T�x

2��0 ðx2jC1 þ C2Þ1=2�;

(13)

where Ujmin;max < 0 are defined as the zeros of the argu-

ment of the square root. In this range of Uj the square root

in the exponent develops an imaginary part. (The comple-ment of the integration region in Eq. (13) provides acorrection to the real part of the potential due to fluctua-tions about the extremal configuration that is not consid-ered here.) Note that in this case U� þ U�Uh; i.e., thebottom of the fluctuating string touches the horizon. Thismay be viewed in the dual gauge theory as a processanalogous to Landau damping of the static color fieldswhich bind the quarks together, leading to the formationof two unbound heavy quarks in the high-temperatureplasma.

On the other hand, at lower temperatures on the order ofthe QCD crossover temperature and below, the dominantprocess should instead correspond to the breakup Q �Q !ðQ �qÞð �QqÞ into color-singlet heavy-light bound states; qstands for a light quark. Such tunneling processes providean exponentially small thermal width of deeply boundstates [12]. It should be clear that the problem of quarktunneling cannot be solved rigorously since it involvesgenuinely nonperturbative QCD dynamics. However, thelarge mass of the heavy quark allows one to use thequasiclassical approximation [12]. For a calculation ofQ �Q ! ðQ �qÞð �QqÞ via the AdS/CFT correspondence seeRef. [13]. Here, too, the temperature must be sufficientlylow to allow for the formation of two new heavy-lightquark bound states. In the gravity description, the finalstate corresponds to two strings connecting the Q branewith the q brane while in our approach they connect to theblack-hole horizon.

The factor 1=�0 in the exponent implies that the leading

order contribution is of orderffiffiffiffi�

p. In the supergravity

approximation � � 1 and, thus, Ij can be computed in

the saddle-point approximation. This gives U ¼ �V 0�=V00�and so

expf�iTVQ �Qg¼Yj

Ij�exp

�� T2��0

�Z

jxj<xc

dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�x2C1�V�þV 02� =2V 00�

q

þ iZjxj>xc

dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU02

c þVðUcÞq ��

;

(14)

where xc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�V� þ V 02� =2V 00� Þ=C1

pif this root is real, and

xc ¼ 0 otherwise. The second contribution in the exponentgives the real part of theQ �Q potential which we drop fromnow on (see Refs. [4,5,7,9]). Performing the integrationover jxj< xc we find

ImVQ �Q ¼ � 1

2ffiffiffi2

p�0

�V 0�2V 00�

� V�V 0�

�; (15)

where we used that U00c ð0Þ ¼ V 0�=2. For N ¼ 4 SYM at

large � the potential entering the NG action is given byVðUÞ ¼ ðU4 �U4

hÞ=R4, where R is the radius of AdS5 andUh ¼ �R2T. This gives

ImVQ �Q ¼ � �

24ffiffiffi2

p ffiffiffiffi�

pT34 � 1

; (16)

whereffiffiffiffi�

p ¼ R2=�0 and � Uh=U� < 1. Note that the

equation above applies only when > 3�1=4 � 0:76; oth-erwise ImVQ �Q ¼ 0 because the solution for xc from above

ceases to exist. Thus, in the vicinity of this point the widthgenerated by the imaginary part of the potential (see be-low) is small compared to the binding energy.The dependence of on L and T can be found from

Eq. (7). At small LT we have LT ¼ b with b ¼2�ð3=4Þ= ffiffiffiffi

�p

�ð1=4Þ � 0:38. On the other hand, when thebottom of the classical string comes too close to thehorizon, � 0:85, the U-shaped configuration used herereceives higher-order corrections [11] and cannot be usedanymore. With � LT, the imaginary part of the potential(16) is smaller than the dominant contribution to the realpart, Eq. (8), by a factor �ðLTÞ4. Here, we consider onlytemperatures such that thermal screening corrections to thereal part of the potential are small; the bound state thenprobes the potential only in the region LT < 1.The imaginary part in Eq. (16) shifts the Bohr energy

level obtained with the Coulomb-like vacuum potential (8),E0 ! E0 � i�; to first order,

�Q �Q � �hc jImVQ �Qjc i ¼ �ffiffiffiffi�

p

48ffiffiffi2

p b

a0

�45

�a0T

b

�4 � 2

�;

(17)

where jc i denotes the unperturbed Coulomb ground state

wave function and a0 ¼ �ð1=4Þ4=2�2ffiffiffiffi�

pmQ is the Bohr

radius. The width decreases with the quark mass and withthe ’t Hooft coupling, approximately as �Q �Q � 1=�m3

Q; it

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increases rapidly with the temperature, �T4. For mQ ¼4:7 GeV, T ¼ 0:3 GeV,

ffiffiffiffi�

p ¼ 3 [14] we obtain �� ’48 MeV.

It is interesting to compare this result to the imaginarypart of the heavy quark potential computed in pQCD:ImVQ �Q ���2

sCFNcTðLTÞ2 logðLTÞ�1 [10]. This is

smaller than ReVQ �Q � �sCF=L by one power of the

coupling and three powers of LT. With L�ð�sCFmQÞ�1, the width decreases less rapidly with the

quark mass than for N ¼ 4 SYM at strong coupling.However, the numerical value of �� is on the order oftens of MeV, similar to what we obtain here.

We suggest that the width computed above is accessibleexperimentally through the suppression of� ! ‘þ‘� pro-cesses in heavy-ion collisions at RHIC or LHC. Neglecting‘‘regeneration’’ of bound states from b and �b quarks in themedium we can estimate the number of � mesons in theplasma at midrapidity, which have not decayed into un-bound b and �b quarks up to time t after the collision, from

dN

dt¼ ���ðTðtÞÞNðtÞ ! NðtÞ ’ N0 exp

��Z

dt��ðtÞ�:

(18)

This solution assumes that ��ðTðtÞÞ is a slowly varyingfunction of time. The initial number of � states may beestimated from the multiplicity in pþ p collisions timesthe number of binary collisions at a given impact parame-ter: N0 ’ NcollN

�pp. Thus, the integrated ‘‘nuclear modifi-

cation factor’’ RAA for the process � ! ‘þ‘� is

approximately given by RAAð� ! ‘þ‘�Þ ’ expð� ���tÞ,where �� denotes a suitable average of �ðTÞ over the life-time of the quark-gluon plasma. Because of the strongtemperature dependence of the width, this average is domi-

nated by the early stage and thus we expect that �� providesan estimate of the initial temperature in heavy-ion colli-

sions via Eq. (17). For t ¼ 5 fm=c and ��� ¼ 48 MeV weobtainRAA ’ 0:3. An experimental estimate for the thermal

quarkonium decay rate �� could be obtained once a statis-tically significant detection of the � ! ‘þ‘� process hasbeen achieved [15].

We thankW. Zajc for useful comments on the preprint ofthis manuscript. J. N. acknowledges support from U.S.-DOE Nuclear Science Grant No. DE-FG02-93ER40764.A. D. gratefully acknowledges support from The CityUniversity of New York through a PSC-CUNY grant andby the Office of Nuclear Physics, U.S.-DOE Grant No. DE-

FG02-09ER41620. J. N. and A.D. also thank GoetheUniversity for their hospitality and the HelmholtzInternational Center for FAIR for support within theLOEWE program.

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