Heavy Quark Potential at Finite-T in AdS/CFT

Yuri KovchegovThe Ohio State University

work done with J. Albacete and A. Taliotis, arXiv:0807.4747 [hep-th]

Outline

AdS/CFT techniques Heavy quark potential at finite-T at strong coupling for N=4

SYM Heavy quark potential at weak coupling at finite-T: and

what we really know there Conclusions

AdS/CFT techniques

AdS/CFT Approach

z

z=0

Our 4dworld

5d (super) gravitylives here in the AdS space

AdS5 space – a 5-dim space with a cosmological constant = -6/L2.(L is the radius of the AdS space.)

5th dimension

222

22 2 dzdxdxdxz

Lds

AdS/CFT Correspondence (Gauge-Gravity Duality)

Large-Nc, large g2 Nc N=4 SYM theory in our 4 space-timedimensions

Weakly coupledsupergravity in 5danti-de Sitter space!

Can solve Einstein equations of supergravity in 5d to learn about energy-momentum tensor in our 4d world in the limit of strong coupling! Can calculate Wilson loops by extremizing string configurations. Can calculate e.v.’s of operators, correlators, etc.

Calculating Wilson loops using AdS/CFT correspondence

Our 4dworld

Wilsonloop

String stretching into the 5th dimension of AdS5.

NGSieW ~

To calculate a Wilson loop, need to i. find the extremal (classical) string configurationii. find the corresponding Nambu-Goto action SNG

iii. then the expectation value of the Wilson loop is given by

Maldacena, ‘98

z

Energy-momentum tensor is dual to the metric in AdS. Using Fefferman-Graham coordinates one can write the metric as

with z the 5th dimension variable and the 4d metric.

Expand near the boundary of the AdS space:

For Minkowski world and with

Holographic renormalization

22

22 ),(~ dzdxdxzxgz

Lds

),(~ zxg

),(~ zxg

de Haro, Skenderis, Solodukhin ‘00

Finite-T medium in AdS

Finite-T static medium in AdS is represented by a black hole solution:

with zh=1/ ( T) the black hole horizon, T – the temperature of gauge theory.

4

44

4

~

222~

2

22

11~

h

h

zzz

z dzxddt

z

Lds

z

black hole

horizonzh

0 our world

Finite-T medium in AdS The black hole metric can be recast into Fefferman-Graham

coordinates:

Here Using holographic renormalization prescription we read off the

energy-momentum tensor of the thermal medium:

222

2

2

22

40

4

40

4

40

4

11

1dzxdd

z

Lds

zz

zz

zz

hzz 20

z

y

x

t

zL

zL

zL

zL

NT C

40

2

40

2

40

2

40

2

2

2

/000

0/00

00/0

000/3

2

Heavy Quark Potential at Finite-Tat Strong Coupling

Heavy Quark Potential At finite temperature the heavy quark potential is defined using a

correlator of two Polyakov loops (in Euclidean time formalism)

Quark and anti-quark can be in the singlet or adjoint (octet) color state. Hence there are two potentials:

We want to find the singlet potential V1(r).

singlet adjoint (octet)

2

)(2)( )1()()0(

1

c

rVc

rV

c N

eNerLL

adj

)(~ 0CNo )/1(~ 2

CNo

NC Counting for Potentials At LO the Wilson loop correlator is zero:

One gluon exchange = o(beta) term. Equating that to zero we obtain:

The adjoint gluon potential is repulsive and NC-suppressed.

2

21

2

)(2)(

)()1()(1

)1()()0(

1

c

adjc

c

rVc

rV

c

N

rVNrV

N

eNerLL

adj

1

)()(

21

cadj N

rVrV

Wilsonloop

Wilsonloop

=0

Heavy Quark Potential Using the rules of AdS/CFT to find the correlator of Polyakov loops

we connect the strings in all possible ways, obtaining the following configurations:

Each configuration is a saddle point in the integral over string coordinates. Summing over the saddle point contributions yields:

2

2 )1()()0(

c

Sc

S

c N

eNerLL

straightNG

hangingNG

black hole horizon

connect toNc D3-branes

Bak, Karch,Yaffe ‘07

NC Counting in AdS Space

There are NC2 configurations of two straight strings, as

each of them can connect to one of the NC different branes:

In black hole metric strings still “remember” which D3 they connect to through Chan-Paton indices.

NC D3-branes

horizon

The Potentials Compare the two formulas:

We conclude that the hanging string configuration yields a singlet potential V1(r).

The two straight strings configuration gives the repulsive adjoint potential Vadj(r).

As Vadj(r)=o(1/NC2) one should expect Sstraight

ren=0 at LO in NC: this is exactly what happens!

2

)(2)( )1()()0(

1

c

rVc

rV

c N

eNerLL

adj

2

2 )1()()0(

c

Sc

S

c N

eNerLL

straightNG

hangingNG

field th’y

AdS

Calculating Singlet Potential

Our 4dworld

Polyakovloops

String stretching into the 5th dimension of AdS5.

/)(1renNGSrV

To calculate singlet potential V1(r), need to i. find the extremal (classical) hanging string configurationii. find the corresponding Nambu-Goto action SNG

iii. then the potential will be given by

z

r

r

0

Euclideantime

Hanging String Configuration We extremize the Nambu-Goto action

to obtain the classical string configuration: We label the string by its maximum

extent in the 5th dimension zmax.

The solution for the string is (Rey et al, hep-th/9803135; Branhuber et al, hep-th/9803137):

Maximum extent zmax is determined by the following equation:

Complex saddle point There are several solutions for zmax , most of them complex!

Two more relevant complex roots for zmax are shown here.

The solutions become complex starting from some minimum separation r=rc.

Complex saddle point

In picking the correct solution/string configuration we demand that the resulting potential is physical. We demand that:

V1(r) becomes Coulomb-like as r→0. In fact it shout map onto T=0 potential calculated by Maldacena in ’98.

As we demand that Im[ V1(r) ] < 0 .

This leaves us with the root denoted by the solid line in the plots.

What to do when string coordinates become complex? Let’s analytically continue into complex domain, similar things are done in the “method of complex trajectories” in quasi-classical QM.

tEtEi ee ]Im[~

Status of the field before this work Before us people did not identify the two string configurations as

corresponding to two different potentials – singlet and adjoint. People would also stop the calculation if string coordinates became

complex. In fact they stop the calculation even earlier, when the energy of the hanging string configuration becomes zero (equal to the energy of the two straight strings configuration).

The resulting heavy quark potential was (Rey et al, hep-th/9803135; Branhuber et al, hep-th/9803137):

Dashed line – their expectation of what the answer should look like;solid line – their answer.

kink?zero potential?

=r

=V(r)

UV regularization and the answer

If one substitutes the string coordinates into the Nambu-Goto action to find the classical NG action, one gets UV divergences due to infinite masses of the quarks.

To renormalize the theory one has to subtract those divergences out. This procedure is defined up to a r-independent constant which may move the resulting V(r) up or down by an overall constant.

We will use the following prescription:

The answer is:

Heavy Quark Potential: Real Part N=4 SYM theory is not confining. At T=0 one only gets Coulomb-like

(~ -1/r) potential.

At finite-T we get:

at small distancesscales as Coulombpotential ~ -1/r

at large distances itfalls off as a power law,proportional to 1/r4 !

No Debye-type screening!

Asymptotics of Re part By exploring the large-r and small-r limits of the full result we get the

following asymptotics:

At small-r we recover Maldacena’s ’98 vacuum result:

At large-r we get an unexpected power-law falloff, instead of usually expected exponential falloff:

Does not contradict any fundamental principles (more later).

Heavy Quark Potential: Real Part

The (real part of the) potential is well-approximated by

with the “screening length”

constrr

r

rrV

30

30

1 )()](Re[

Tr

702.2

0

Heavy Quark Potential: Imaginary Part

The potential becomes absorptive at large separations: it develops an imaginary part. This is due to color singlet state easily decaying into acolor-octet (adjoint) configuration.

at large distances scales ~r

cf. perturbative QCDcalculations byM. Laine et al,hep-ph/0611300; N. Brambilla et al,arXiv:0804.0993.

Asymptotics of Im part

At large-r the Im part of the potential is

Makes sense: the wider part the quarks are the more likely the singlet state to decay into an octet (adjoint) state.

Due to Im part the color-singlet state is a metastable state which will of course decay into an octet state with a high probability.

One can think of Im part of V(r) as meson decay width in a thermal medium.

Adjoint potential

To find the adjoint potential consider two straight strings:

The LO part is r-independent and after renormalization gives Vadj(r)=0, in agreement with expected 1/NC

2 suppression.

NLO graviton exchange corrections are o(1/NC2) and can be

calculated (see Bak, Karch,Yaffe ‘07) giving Vadj(r).

CN

1

CN

1

graviton

Heavy Quark Potential at Finite-Tat Weak Coupling

Perturbative Potential

The standard perturbative calculationyields:

),0(~)(

0002

3

qqq

eqdrV

rqi

Usually people approximate (exact for Abelian theories)

and get

2000 ),0( Dmqq

r

erV

rmD

~)(

Perturbative Potential

However, in non-Abelian theories the static self-energy is not a constant:

Keeping the linear term in self-energy yields (C. Gale, J. Kapusta ‘87):

Note that one gets the same power as at strong coupling in AdS!

||#),0( 22000 qTgmqq D

43

1~)(

rTrV

Perturbative Potential

A word of caution: the linear term becomes important when it is comparable to q2:

This happens at and hence

We are dealing with ultrasoft (magnetic) modes, which are non-perturbative. Hence we should not trust either exponential and power-law screening. It is curious though that is we take perturbative expression “as is” we get a power-law falloff.

),0(~)(

0002

3

qqq

eqdrV

rqi

qTgq 22 ~ Tgq 2~

Conclusions We have identified the hanging string and two straight strings

configurations with the singlet and adjoint heavy quark potentials.

We have used the complex-string worldsheet technique to find the singlet heavy quark potential in the strongly coupled SYM theory at finite temperature.

The resulting potential is smooth, has no kinks, with a negative-definite Re part, and has a non-zero Im part giving the decay width for the meson states.

The Re part of the potential at large separations falls off as 1/r4: an interesting alternative to exponential Debye screening.

Apparently there is no consensus on what the screened potential is at weak coupling.