Heavy Quarkonium Melting in Large N Thermal QCD

# Heavy Quarkonium Melting in Large N Thermal QCD

Mohammed Mia, Keshav Dasgupta, Charles Gale, Sangyong Jeon
Department of Physics, McGill University,
3600 University Street, Montréal QC, Canada H3A 2T8
November 2009
###### Abstract:

Large QCD is mostly governed by planar diagrams and should show linear confinement when these diagrams are suitably summed. The linear confinement of quarks in a class of these theories using gravity duals that capture the logarithmic runnings of the coupling constants in the IR and strongly coupled asymptotic conformal behavior in the UV was studied in our previous work. We also extended the theories to high temperatures and argued the possibilities of meltings and suppressions of heavy quarkonium states. In this paper we give a formal proof of melting using very generic choices of UV completions, and point out some subtleties associated with meltings in generic large theories. Our proof requires only the existence of well defined UV behaviors that are devoid of Landau poles and UV divergences of the Wilson loops, allowing degrees of freedom to increase monotonously with energy scale. We determine the melting temperatures of heavy quarkonium states, which could suggest the presence of deconfinement phase transitions in these theories.

## 1 Introduction

Large QCD in dimension is a relatively simpler theory than its finite counterpart, owing in part to the suppressions of the non-planar diagrams, but its solution is still a challenge. For example its not clear how to add up all the planar diagrams and argue for linear confinement of fundamental quarks. Although in terms of mesonic and glue-ball degrees of freedom the theory may look free, this is an oversimplification. Owing to the existence of a Hagedorn temperature and, at a more fundamental level, quark degrees of freedom, the system is still complex with no simple way of computing, for example, the current-current correlator or the master field [1].

In principle, gauge-gravity duality provides a way to compute - or at least allow some analytic control on - some of these quantities. However the problem is that to restrict everyting to the supergravity level, where we can have more analytic control, the gauge theory should be at strong ’tHooft coupling. This cannot happen for large QCD that is asymptotially free (although there could be a full string theory dual description). If one relaxes that condition and looks for strongly coupled asymptotic conformal behavior then one can construct a supergravity dual for a class of these theories.

In a previous paper [3], which was a continuation of earlier work [2], a new supergravity background that captures the logarithmic runnings of the coupling constants of a particular class of large QCD in the far IR and the strongly coupled asymptotic conformal behavior in the far UV was constructed. In the intermediate energy scales, our dual gravitational background captures the interpolating behavior of the beta function.

In [3] it was argued that such a geometry would consist of three regions termed region 1, 2 and 3 that would capture the IR, the intermediate scale, and the UV respectively. The supergravity solution was constructed using fluxes sourced by number of D3 branes, number of D5 branes and anti-branes respectively while taking the back reaction of number of seven branes and number of anti seven branes. The metric in all three regions can be written in the following form [3]

 ds2=1√h[−g1dt2+dx2+dy2+dz2]+√h[g−12grrdr2+gmndxmdxn] (1)

with being the black-hole factors and being the warp factor depending on all the internal coordinates (). To zeroth order in and we have our usual relations:

 h[0] = L4r4,    g[0]=1−r4hr4,    g[0]rr=1,    g[0]mndxmdxn=ds2T11 g[0]1=g[0]2≡g=1−r4hr4,      L4≡gsNα′2 (2)

where is the base of a six dimensional conifold and it has the topology of , where and are three-sphere and two-sphere. Here is the string coupling, is the AdS throat radius and is the black hole horizon. The above metric is that of a ten dimensional geometry with coordinates describing a five dimensional non compact space while internal coordinates () describe a five dimensional compact space.

At higher orders in , the warp factor, the black hole factors111This also means that the black hole horizon is no longer the surface and the equation for the surface may be more involved. and the internal metric get modified because of the back-reactions from the seven-branes, three form fluxes and the localized sources we embed. We can write this as:

 h = h[0] + h[1],   grr = g[0]rr + g[1]rr,   gmn = g[0]mn + g[1]mn,   gi=g[0]i+g[1]i (3)

where the superscripts denote the order of and .

In regions 1 and 2, the warp factor has logrithmic dependence while in region 3, it can be expanded exclusively as a power series in . In terms coordinate, we have the following Taylor series expansion for the warp factor :

 1√h=Anun−2 (4)

where are Taylor coefficients and in general can be functions of internal coordinates. Region 3 was taken to be large enough so that the Nambu-Goto string would lie completely inside it. The computation of the potential at zero temperature revealed that at short distances the potential should be dominated by the Coulomb term222In deriving the Coulomb potential, we have set AdS throat radius and string tension for convenience, and this is only a redefinition of units. By restoring units, one obtains , as expected [4].[3]:

 VQ¯Q = −0.236d + O(d) (5)

where the numerical value and the sign of the first term are fixed naturally by the dual background. It is interesting to note that this value is of the same order of magnitude as the one derived from QCD lattice simulations [5]. For large distances one expects the potential to be dominated by the linear term [3]:

 VQ¯Q = (Anxnmaxπx2max) d (6)

where is the maximum value for the depth of the U shaped string denoted by and it is a solution of the constraint equation (see section 2 and also [3] for details):

 12(m+1)Am+3xm+3max = 1 (7)

The constraint equation is obtained by demanding that the distance between the quarks and the potential energy of the quark pair is real. For AdS space, interquark distance and potential is always real for any value of , and hence do not exist, as there in no constraining equation.

At high temperatures and density, medium effects should screen the interaction between the heavy quark the anti-quark. The resulting effective potential between the quark anti-quark pairs separated by a distance at temperature can then be expressed succinctly in terms of the free energy , which generically takes the following form:

 F(d,T)=σd fs(d,T)−αdfc(d,T) (8)

where is the string tension, is the gauge coupling and and are the screening functions333We expect the screening functions to equal identity when the temperature goes to zero, yielding the zero temperature Cornell potential. (see for example [6] and references therein). For the quark and the anti-quark pair kept at respective positions of and , we expect the Wilson line to be related to the free energy through

 exp[−F(d,T)T] = ⟨W†(+d2)W(−d2)⟩⟨W†(+d2)⟩⟨W(−d2)⟩ (9)

In terms of Wilson loop, the free energy Eq.(8) is now related to the renormalised Nambu-Goto action for the string on a background with a black-hole444There is a large body of literature on the subject where quark anti-quark potential has been computed using various different approaches like [7]. Although the potential Eq.(10) that we get matches well with other results, we have an additional constraint: Eq.(1), from the background RG flow. This additional constraint, which cannot be seen from an AdS or an AdS with an IR cut-off dual, will have non-trivial consequences for melting that we will discuss in section 2.. One may also note that the final theory is not three-dimensional, but four-dimensional and compactified on a circle in Euclideanised version.

Using Eq.(9) and the identification of the Wilson loop to the Nambu-Goto action, the free energy (or equivalently the potential) between the heavy quark and the anti-quark at non-zero temperatures can be deduced. For large distances the potential is [3]

 VQ¯Q =  ⎷1−y4maxu4h(Anynmaxπy2max) d (10)

where is the maximum value for the depth of the U shaped string in the presence of a black hole. It is given by solving the constraint equation:

 12(m+1)Am+3ym+3max+1j!j−1∏k=0(k−12)(y4maxu4h)j[Alylmax(l2+2j−1)] = 1

which arises by demanding that inter quark separation and free energy is always real [3]. Note that and for zero temperature, . Thus at zero temperature, and (1) reduces to (7).

At this point, it is important to define in a more precise manner what exactly is meant by melting. In this work, “melting” is meant to quantify the disappearance of the linear portion of the quark-antiquark potential, at a given length scale. There is therefore a qualitative difference between this behavior and the actual dissolution of quarkonium bound states: The robustness of the pairs will actually depend on where their energy sits in the potential profile. Later studies could also involve a more precise characterization through a study of the temperature dependence of quarkonium spectral densities, for example.

Going back to Eq.(10), one might think that the melting temperature is given by the condition that is a solution of Eq.(1) so that the coefficient of in Eq.(10) vanishes. Actually, this turned out not to be the case. As shown below, Eq.(1) can never have a solution at . Hence, the deconfinement in our case does not mean a complete absence of the linear potential. Rather, it means that the linear potential ceases to have an infinite range and as temperature increases, the range of the linear potential quickly becomes short provided that Eq.(1) allows a real positive solution lying in region 3. It is not clear this would always be the case, so Eq.(1) requires a more detailed investigation. In the following section i.e sec. 2, we will carefully analyse Eq.(1) for generic choices of warp factors or more appropriately, generic UV completions that have no Landau poles or UV divergences of the Wilson loops, and give a proof of quarkonium melting for this class of theories. Section 3 contains a detailed numerical analysis that will allow us to find the melting points, and could also suggest the presence of a deconfinement phase transition in these theories. We conclude with a short discussion.

## 2 Proof of the existence of a melting point

Our cascading picture of renormalization group flow demands that in the region 3, the effective number of colors grows as grows. The number of colors at any scale in the region 3 is given by . For the analysis given here, it is simpler to define and use

 F(u)≡u2√h=√NL2√Neff=1L2√1+alul (12)

instead of . The coefficients appearing in Eqs. (4), (7) and (1) are related to by ; and is the warp factor. In terms of , the condition that is a decreasing function of becomes

 F′(u)>0 (13)

Combining Eqs.(12) and (13) yields the following condition

 F(u)>1L2 (14)

From now on, and as mentioned earlier, the value of is set to so that .

### 2.1 Zero temperature

Let be the maximum value of for the string between the quark and the anti-quark. Then the relationship between and the distance between the quark and the anti-quark is given by [3]

 d(umax)=2umaxF(umax)∫1ϵ0dvv2√Gmummaxvm(F(umaxv))2⎡⎣1−v4(F(umax)F(umaxv))2⎤⎦−1/2 (15)

Two conditions must be met before asserting that this expression represents the physical distance between a quark and an anti-quark in vacuum. One obvious condition is that the integral is real. This is guaranteed if for all :

 W(v|umax)≡v2(F(umax)F(umaxv))≤1 (16)

Another condition is that the potential between the quark and the anti-quark must be long ranged. That is, must range from 0 to as varies from 0 to some finite value, say . Since , the only way to satisfy these conditions is via the (sufficiently fast) vanishing of the square-root in Eq.(15) as at .

For most , vanishes only linearly as approaches 1. In this case, is finite since the singularity in the integrand behaves like and hence integrable. To make diverge at , must vanish quadratically as approaches 1 so that the integrand is sufficiently singular, . Therefore, the function must have a maximum at .

To determine the value of , consider

 W′(v|xmax) = 2v(F(xmax)F(xmaxv))(1−(xmaxv)F′(xmaxv)2F(xmaxv)) (17)

For this to vanish at , must be the smallest positive solution of

 xF′(x)−2F(x)=0 (18)

With the definition , one can easily show that this is equivalent to the condition (7) which was originally derived in [3]. The allowed range of is then

 0≤umax≤xmax (19)

and within this range, varies from 0 to . How it varies will depend on the values of as well as .

### 2.2 Finite temperature

At finite temperature, the relation between and the distance between the quark and the anti-quark is obtained by replacing with in Eq.(15):

 dT(umax) = 2umax√1−u4max/u4hF(umax)∫1ϵ0dvv2√Dmummaxvm(1−v4u4max/u4h)(F(umaxv))2 (20) ×⎡⎣1−v4(1−u4max/u4h)(1−v4u4max/u4h)(F(umax)F(umaxv))2⎤⎦−1/2

The explicit factor of makes vanish at as in the case. As approaches , the integral near behaves like

 dT(umax) ∼ ∫10dv√1−u4max/u4h√(1−v)(1−vumax/uh) (21)

which indicates that goes to 0 as approaches . Hence, at both and , vanishes. Since is positive in general, there has to be a maximum between and . Whether the maximum value of is infinite as in the case depends on the temperature (equivalently, ) as we now show.

The fact that the physical distance needs to be real yields the following condition. For all ,

 WT(v|umax)≡v2(F(umax)F(umaxv)) ⎷1−u4max/u4h1−u4maxv4/u4h≤1 (22)

Taking the derivative gives

 W′T(v|umax) = (1−umax/u4h)1/2(1−u4maxv4/u4h)3/2vF(umax)F(umaxv) ×[−(umaxv)(1−(umaxv/uh)4)F′(umaxv)+2F(umaxv)]

Similarly to the case, can have an infinite range if the derivative vanishes at for a certain value of , say . This value of is determined by the smallest positive solution of the following equation

 yF′(y)−2F(y)=(y/uh)4yF′(y) (24)

which then forces to vanish. Note that the left hand side is the same as the zero temperature condition, Eq.(18). The right hand side is the temperature dependent part. Using the facts that:

 j−1∏k=0(k−1/2)=−(2j−3)!!/2j,   ∞∑j=1xj(2j−3)!!/2jj!=1−√1−x, (25)

it can be readily shown that Eq.(24) is equivalent to Eq.(1) as long as . It is also clear that cannot be a solution of Eq.(24) because at , the equation reduces to which is inconsistent with the fact that .

Recall that and in the region 3 and we assume that the equation has a real positive solution . Hence as increases from 0 towards , the left hand side of Eq.(24) increases from while the right hand side increases from 0. The left hand side reaches 0 when which is the point where the distance at becomes infinite. At this point the right hand side of Eq.(24) is positive and has the value . Hence the solution of Eq.(24), if it exists, must be larger than .

Consider first low enough temperatures so that . For these low temperatures, Eq.(24) will have a solution, as the right hand side will be still small around . This then implies that the linear potential at low temperature will have an infinite range if the zero temperature potential has an infinite range.

To show that the infinite range potential cannot be maintained at all temperatures, let . When the left hand side vanishes at , the right hand side is which is positive and finite. For , the left hand side () may become postive, but it is always smaller than since is always positive. But for the same , the right hand side () is always positive and necessarily larger than since . Hence, Eq.(24) cannot have a real and positive solution when . Therefore between and , there must be a point when Eq.(24) cease to have a positive solution.

When Eq.(24) has no solution, then the expression for , (20) will not diverge for any within . Furthermore, since the expression vanishes at both ends, there must be a maximum at a non-zero . When the distance between the quark and the anti-quark is greater than this maximum distance, there can no longer be a string connecting the quark and the anti-quark.

## 3 Numerical analyses of melting temperatures

After discussing the most general choice for warp factors that give rise to and consequently linear potential, specific examples of geometries that may arise as solutions to Einstein’s equation will be considered, starting with the following ansatz for the metric:

 ds2 = −g√hdt2+1√h(dx2+dy2+dz2)+√hu2(Hgu2du2+ds2M5) (26) ≡ −g√hdt2+1√h(dx2+dy2+dz2)+√hu2˜gmndxmdxn

where , and ; is the compact five dimensional manifold parametrised by coordinates and can be thought of as a perturbation over . Here is the boundary and is the horizon. As discussed in earlier work [3], the above metric arises in region 3 of [3] when one considers the running of axio-dilaton , brane local action and fluxes due to anti five-branes on a geometry that deviates from the IR OKS-BH geometry from the backreactions of the above sources. The three-form fluxes sourced by () anti-branes are proportinal to for some positive (see [3] for details about ), where the function as and as . With the coordinate , there is another function: , that also has somewhat similar behaviour as and may allow us to have a better analytic control on the background. With such a choice of , the total three from flux is proportional to with

 M(u)≡M[1−k(u)]=M[1−exp(−uA)] (27)

where is the number of bi-fundamental flavors. Thus three-form fluxes are decaying fast as and, as shown in [3], the seven-branes could be arranged such that the axio-dilaton behaves typically as . This means that from the behavior of the internal Riemann tensor one may conclude that the internal metric behaves as where and are all positive and could be positive or negative depending on the precise background informations.

From the above discussions it should be clear that taking the three-forms and world-volume gauge fluxes to be exponentially decaying in the IR (but axio-dilaton to be suppressed only as ) should solve all the equations of motion, giving the following behavior for the warp factor and the internal metric in (26)555See also the interesting works of [8] where exponential warp factors have been chosen.:

 h = L4u4exp(−αu˜α),      H = exp(βu˜β) (28)

where we are taking to be all positives with to be functions of internal coordinates () and to be the asymptotic AdS throat radius666Note that in (28) could be considered negative so that would be decaying to zero in the IR. However since region 3 doesn’t extend to the IR we don’t have to worry about the far IR behavior of Eq.(28)..

Motivated by the above arguments, we will consider Nambu-Goto action of the string in the geometry with and at temperatures and respectively in Eq.(28). As in [3, 2] we consider mappings , which are points in the internal space, to lie on the slice:

 θ1 = θ2 = π,      ϕi = 0,      ψ = 0 (29)

so that on this slice are fixed and we set it to for both choices . (Such a choice of slice will also help us to ignore the three-form contributions to the Wilson loop.) With these fixed choices for the warp factors, we plot the interquark separation as a function of in figures 1 and 2 for various values of .

Note that for both choices of warp factors, for low enough temperatures, there exist where . As the temperature is increased, increases modestly. On the other hand from figure 1, one sees that when there exists a which is finite. This means for interquark distance , there is no string configuration with boundary condition implying that the string attaching the quarks breaks and we have two free partons for . Thus we can interpret to be a “screening length”. From figure 2 we observe similar behaviour but now exists for .

In figure 3, as a function of is plotted. We note that for a small change in the temperature near (or near equivalently) there is a sharp decrease in screening length , but for , , the screening length does not change much. In fact behaves as (where and are constants) which in turn could be an indicative of a phase transition near for i.e the two choices of warp factor.

Finally we plot the potential as a function of in figures 4 and 5 for the two choices of warp factor. For in figure 4 and in figure 5, we have potential energies linearly increasing with an arbritrarily large increment of the interquark separations. Thus we have linear confinement of quarks for large distances and small enough temperatures. For , or , there exists a and for all distances there are no Nambu-Goto actions, , for the string attaching both the quarks. This means that we have free quarks and is constant for . Of course looking at figures 4 and 5 one shouldn’t conclude that the free energy stops abruptly. What happens for those two cases is that the string joining the quarks breaks, and then the free energy is given by the sum of the energies of the two strings (from the tips of the seven-branes to the black-hole horizon) and the total energies of the small fluctuations on the world-volume of the strings. The latter contributions are non-trivial to compute and we will not address these in any more detail here, but energy conservations should tell us how to extrapolate the curves in figures 4 and 5, beyond the points where the string breaks, for all . Of course after sufficiently long time the two strings would dissipate their energies associated with their world-volume fluctuations and settle down to their lowest energy states.

Observe that for a wide range of temperatures , the potential and thus the free energy hardly changes. But near a narrow range of temperatures (where ), free energy changes significantly. For figure 5 the change is more abrupt than figure 4. This means as we go for bigger values of , the change in free energy is sharper.

In figure 6, we plot the slope of the linear potential as a function of . Again for a wide range of temperatures, there are no significant changes in the slope but near , the change is more dramatic: the slope decreases sharply, indicating again the possibility of a phase transition near . As we noticed before, here too bigger exponent gives a sharper decline in the slope hinting that when , the transition would be more manifest.

To conclude, the above numerical analyses suggest the presence of a deconfinement transition, where for a narrow range of temperatures the free energy of pair shows a sharp decline. Interestingly, changing the powers of in the exponential changes the range of only by a small amount. So effectively lies in the range . Putting back units, and defining the boundary temperature777See sec. (3.1) of [2] for details. as , our analyses reveal:

 0.91L2 ≤ Tc ≤ 1.06L2 (30)

which is the range of the melting temperatures in these class of theories for heavy quarkonium states. Since the temperatures at both ends do not differ very much, this tells us that the melting temperature is inversely related to the asymptotic AdS radius in large thermal QCD.

## 4 Discussions and conclusions

In this work we attempted a small step towards the understanding of the melting temperatures of heavy quarkonium states in a class of large thermal QCD that have well defined UV behaviors without Landau Poles and UV divergences of the Wilson loops. Our analyses reveal two interconnecting results: any large gauge theory with fundamental quarks () that is away from conformality with atmost one UV fixed point888This also means a good UV behavior with degrees of freedom increasing monotonously with energy scales. should always have a mass gap and, consequently at a certain temperature , the heaviest quarkonium states in such a theory should show melting. Such a conclusion seems to be true for generic non-supersymmetric theories, and it would be interesting to extend this to the supersymmetric cases.

The melting temperatures999Note that their dependences on the ’tHooft coupling are consistent with what we would expect. that we get for various UV completions seem not to differ too much from each other. This may mean that there is some underlying universal behavior of heavy quarkonium states in large theories with good UV behaviors at strong ’tHooft couplings. An interesting related study would be that of the behavior at weak ’tHooft couplings. There, there are also Hagedorn states that would come in because of the unsuppressed string modes. Details on this will be presented elsewhere.

Acknowledgements

Its our pleasure to thank Sumit Das, Juan Maldacena, Guy Moore and Mark Van-Raamsdonk for helpful comments. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.

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