Correlations of scales in Correlations of scales in AdS/QCDAdS/QCD

S.Dubynskiy, M. Voloshin, A.G. hepth S.Dubynskiy, M. Voloshin, A.G. hepth 0804.22440804.2244

V.Zakharov, A.G. to appearV.Zakharov, A.G. to appear

Question - How two colorless objects Question - How two colorless objects correlate if they have very different sizes?correlate if they have very different sizes?

Motivation – “experimental surprise ” andMotivation – “experimental surprise ” and

“ “lattice puzzle” lattice puzzle”

Experiment (Belle) – resonances near the Experiment (Belle) – resonances near the open charm threshold with unusual decay open charm threshold with unusual decay modes . The suggestion (Voloshin 2007)- modes . The suggestion (Voloshin 2007)- these resonances are the bound states of these resonances are the bound states of the charmonium “inside” the excited light the charmonium “inside” the excited light meson. The only decay modes of these meson. The only decay modes of these resonances – charmonium and light resonances – charmonium and light mesonsmesons

Lattice (2007,Boiko et.al.) – investigation Lattice (2007,Boiko et.al.) – investigation of the internal structure of the QCD string. of the internal structure of the QCD string. Technically- investigation of the correlatorTechnically- investigation of the correlator

of two circular Wilson loops in Euclidean of two circular Wilson loops in Euclidean space <W(r)W(R)> when R is much largerspace <W(r)W(R)> when R is much larger

then r. The unexpected dependence on rthen r. The unexpected dependence on r

is found numerically. The investigation is found numerically. The investigation concerns the broading of the string shape concerns the broading of the string shape and the correlator behaves as r^2. Being and the correlator behaves as r^2. Being translated at the string width it would translated at the string width it would mean that the string becomes very thin.mean that the string becomes very thin.

We shall try to handle both problems We shall try to handle both problems within AdS/QCD approachwithin AdS/QCD approach

Consider the simplest soft wall model Consider the simplest soft wall model (Karch et.al 2006) with nontrivial dilaton(Karch et.al 2006) with nontrivial dilaton

The heavy meson is considered asThe heavy meson is considered as localized near UV boundary z=0 withlocalized near UV boundary z=0 with the characteristic scale of z~1/m. On the characteristic scale of z~1/m. On

thethe other hand the light meson wave other hand the light meson wave

function is mainly localized at the IR function is mainly localized at the IR region and has nontrivial z dependence. region and has nontrivial z dependence. It can be treated in terms of modes of It can be treated in terms of modes of 5D fields5D fields

Physics behind interaction- heavy Physics behind interaction- heavy quark contribution to the van der Waals quark contribution to the van der Waals type interaction due to the chromo-type interaction due to the chromo-polarizability of the heavy mesonpolarizability of the heavy meson

where E - chromoelectric field. The nextwhere E - chromoelectric field. The next

step- relate Hstep- relate Heff eff to the conformal to the conformal anomalyanomaly

We replace the electric interaction with the sum of the electric and magnetic terms. This corresponds to a conservative treatment of the problem of the boundstates because of the sign-definite magnetic contribution.

That is the interaction of the light meson X with That is the interaction of the light meson X with heavy meson is due to the dilaton exchange. The heavy meson is due to the dilaton exchange. The light meson light meson

in the model is described by the eigenvalue in the model is described by the eigenvalue problemproblem

where S – meson spin, and

The spectrum has Regge behavior

The heavy meson induces the potential The heavy meson induces the potential for the light meson modes due to the for the light meson modes due to the dialton exchange in the bulk and the 4D dialton exchange in the bulk and the 4D eigenvalue problem for energy w readseigenvalue problem for energy w reads

which involves three-dimensional which involves three-dimensional LaplasianLaplasian

and potential is defined by the dilaton and potential is defined by the dilaton bulk-boundary propagator bulk-boundary propagator

From the dilaton propagator one getsFrom the dilaton propagator one gets

The function f(z) can be found from the The function f(z) can be found from the relationrelation

which yields f(z)= which yields f(z)=

And c is determined by the And c is determined by the polarizability.polarizability.

The problem can be solved by The problem can be solved by variational procedure with the probe variational procedure with the probe functionfunction

Which obeys the equation Which obeys the equation

With the potential With the potential

Assuming the scale factor Assuming the scale factor

the bound state emerges starting from S=4 - the bound state emerges starting from S=4 - numerical calculation.numerical calculation.

However likely the polarizability can However likely the polarizability can considerably larger decreasing the value considerably larger decreasing the value of S.of S.

Experimentally there is no decay ofExperimentally there is no decay of

Hadro-quarkonium into D-mesons. Hadro-quarkonium into D-mesons. Why?Why?

Consider the brane realization of this Consider the brane realization of this statestate

heavy brane

Light brane

horizon

Final state for the decay into two D mesons

This is the stringy tunneling process This is the stringy tunneling process however there are strings in the however there are strings in the initial state, that is initial state, that is INDUCED INDUCED tunneling. Moreover it turns out to be tunneling. Moreover it turns out to be induced tunneling near the top of the induced tunneling near the top of the barrierbarrier

Effective potential for the tunneling process

V(r)

r

Horizontal slice of the Euclidean bounce solution corresponding to the induces decay of the hadro-quarkonium

Euclidean time direction

The process can be also treated as the The process can be also treated as the ionization of the heavy meson in the ionization of the heavy meson in the external field. Both approaches yieldexternal field. Both approaches yield

the same answer for the probabilitythe same answer for the probability

Hence this process is suppresed exponentially indeed

Correlator of two Wilson Correlator of two Wilson loopsloops

Consider the Euclidean correlatorConsider the Euclidean correlator

and assume very different radii. What and assume very different radii. What is the connectness mechanism? Lattice is the connectness mechanism? Lattice tells the unexpected dependence on tells the unexpected dependence on the small radius (2007)the small radius (2007)

The natural expectation was The natural expectation was due to the general arguments concerningdue to the general arguments concerningthe scales of fields in the flux tubethe scales of fields in the flux tube

What AdS/QCD yields for this correlator at strong coupling?What AdS/QCD yields for this correlator at strong coupling?Naively one could expect string worldsheetNaively one could expect string worldsheetwith two boundaries at the Wilson loops.with two boundaries at the Wilson loops.No such minimal surface if radii are No such minimal surface if radii are very different (Olesen-Zarembo 2000).very different (Olesen-Zarembo 2000).

The are two possibilities insteadThe are two possibilities instead

1.1. Exchange by SUGRA mode-dilatonExchange by SUGRA mode-dilaton2. ‘Complex time” solution for the string2. ‘Complex time” solution for the string

Mode exchange. Consider expansionMode exchange. Consider expansion

Immediately we get Immediately we get

That is the dilaton exchange result agrees That is the dilaton exchange result agrees with the common expectations. Is there with the common expectations. Is there any place for the quadratic dependence?any place for the quadratic dependence?

Consider the solution to the stringy equationConsider the solution to the stringy equation

of motion in the “of motion in the “complex time -zcomplex time -z”. The ”. The

solution looks as two “Euclidean regions”solution looks as two “Euclidean regions”

with real actions and the ‘Minkowskianwith real actions and the ‘Minkowskian

region” between with the imaginary action. region” between with the imaginary action. This resembles the QM situation with theThis resembles the QM situation with the

resonant tunneling when penetrationresonant tunneling when penetration

of the particle through of the particle through TWOTWO barriers is of barriers is of order 1 order 1

z

The analogue in QM looks as follows

If the energy of the particle coincides withIf the energy of the particle coincides with

the metastable level between two barriersthe metastable level between two barriers

then the resonant tunneling happens. It then the resonant tunneling happens. It corresponds to the infinite oscillations corresponds to the infinite oscillations

between two barriers. In the case of between two barriers. In the case of extended objects the resonant tunneling extended objects the resonant tunneling was applied to the production at was applied to the production at thresholdthreshold

problem (Voloshin-A.G. 93). In the problem (Voloshin-A.G. 93). In the Minkowski region- oscillating closedMinkowski region- oscillating closed

string. The resonance condition can be string. The resonance condition can be formulated in WKB approximation formulated in WKB approximation

The solution for the string in AdS The solution for the string in AdS consists from three piecesconsists from three pieces

The total action isThe total action is

The resonance condition reduces to The resonance condition reduces to

And taking into account the string And taking into account the string tensiontension

in the strong coupling regimein the strong coupling regime

At the resonance condition the correlatorAt the resonance condition the correlatorof two Wilson loops becomes large and has of two Wilson loops becomes large and has

nontrivial dependence on the small radius.nontrivial dependence on the small radius.

ProblemsProblems1.It is not clear if one can neglect1.It is not clear if one can neglectthe gravitational emission of the string the gravitational emission of the string during the oscillations in the Minkowski during the oscillations in the Minkowski

region. region. 2. Unusual dependence on the 2. Unusual dependence on the coupling constant. Probably the runningcoupling constant. Probably the runningstring tension could help. string tension could help. ……....

There are nontrivial correlators of the There are nontrivial correlators of the composite Wilson loopscomposite Wilson loops

<W(D,R)W(D,r)><W(D,R)W(D,r)>

Where each dyonic Wilson loop Where each dyonic Wilson loop consistsconsists

of arcs of magnetic and electric of arcs of magnetic and electric components. The correlator of such components. The correlator of such composite Wilson loops is saturated composite Wilson loops is saturated by the by the

worldsheet of the dyonic string at any worldsheet of the dyonic string at any

radii.radii.

M

E

D

The horizontal slice of the configuration of two composite Wilson loops involving magnetic(M), electric(E) and dyonic(D) string worldsheets. In the Minkowski case it corresponds to the interaction of E and M degrees of freedom,(Minahan 98)

ConclusionsConclusions

In the Minkowski geometry correlation of In the Minkowski geometry correlation of “IR” and “UV’ scales provides the “IR” and “UV’ scales provides the possibility for the exotic bound state- possibility for the exotic bound state- hadro-quarkonium to existhadro-quarkonium to exist

In the Euclidean geometry the exchange of In the Euclidean geometry the exchange of SUGRA modes - no surprisesSUGRA modes - no surprises

The possibility of the resonant tunnelingThe possibility of the resonant tunneling

in the Euclidean case - interesting in the Euclidean case - interesting nonperturbativenonperturbative IR/UV mixing IR/UV mixing

At the resonance condition the correlatorAt the resonance condition the correlatorof two Wilson loops becomes large and has of two Wilson loops becomes large and has

nontrivial dependence on the small radius.nontrivial dependence on the small radius.

ProblemsProblems1.It is not clear if one can neglect1.It is not clear if one can neglectthe gravitational emission of the string the gravitational emission of the string during the oscillations in the Minkowskian during the oscillations in the Minkowskian

region. region. 2. Unusual dependence on the 2. Unusual dependence on the coupling constant.Probably the runningcoupling constant.Probably the runningstring tension could help. string tension could help. ……....