1 Introduction and summary

ITP-UU-10/39

DFTT 27/2010

D3-D7 Quark-Gluon Plasmas at Finite Baryon Density

Francesco Bigazzi , Aldo L. Cotrone , Javier Mas ,

Daniel Mayerson , Javier Tarrío .

Dipartimento di Fisica e Astronomia, Universitá di Firenze and INFN Sezione di Firenze; Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy.

Dipartimento di Fisica, Universitá di Torino and INFN Sezione di Torino;

Via P. Giuria 1, I-10125 Torino, Italy.

Departamento de Física de Partículas, Universidade de Santiago de Compostela and Instituto Galego de Física de Altas Enerxías (IGFAE); E-15782, Santiago de Compostela, Spain.

Institute for theoretical physics, K.U. Leuven; Celestijnenlaan 200D, B-3001 Leuven, Belgium.

Institute for Theoretical Physics, Universiteit Utrecht, 3584 CE, Utrecht, The Netherlands.

bigazzi@fi.infn.it, cotrone@to.infn.it, javier.mas@usc.es, drm56@cam.ac.uk, l.j.tarriobarreiro@uu.nl

Abstract

We present the string dual to SYM, coupled to massless fundamental flavors, at finite temperature and baryon density. The solution is determined by two dimensionless parameters, both depending on the ’t Hooft coupling at the scale set by the temperature : , weighting the backreaction of the flavor fields and , where is the baryon density. For small values of these two parameters the solution is given analytically up to second order. We study the thermodynamics of the system in the canonical and grand-canonical ensembles. We then analyze the energy loss of partons moving through the plasma, computing the jet quenching parameter and studying its dependence on the baryon density. Finally, we analyze certain “optical” properties of the plasma. The whole setup is generalized to non abelian strongly coupled plasmas engineered on D3-D7 systems with D3-branes placed at the tip of a generic singular Calabi-Yau cone. In all the cases, fundamental matter fields are introduced by means of homogeneously smeared D7-branes and the flavor symmetry group is thus a product of abelian factors.

1 Introduction and summary

Heavy ion collision experiments at RHIC and LHC allow us to explore a relevant corner of the QCD phase diagram (high temperature and relatively small baryon chemical potential), where the theory is expected to be deconfined. Both the results collected during the ten-year run of RHIC [1] and the preliminary ones at LHC [2] actually indicate that a quark-gluon “fireball” is formed and behaves like a strongly coupled system: a liquid with very small viscosity over entropy density ratio. Holographic methods provide interesting tools to analyze these kind of systems. The simplest and best studied example is the conformal SYM plasma which, unexpectedly, has proven to share some properties with the QCD one. This fact has stimulated further research works with the aim of refining this master holographic model, for example by adding fundamental matter fields. The latter has been performed mainly in the quenched approximation.

In [3] some of the authors have presented a ten dimensional black-hole solution dual to the non conformal plasma of SYM coupled to massless flavors.111All the hydrodynamic transport coefficients of the model were derived in [4]. The latter were introduced by means of homogeneously smeared D7-branes [5, 6, 7], extended along the radial direction up to the black hole horizon. The smearing reduces the flavor symmetry group to a product of abelian factors and allows a simple way to account for the backreaction of the D7-branes and thus to explore the “unquenched” regime in the dual field theory.222For other holographic studies of thermal unquenched flavors, see [8]. The analysis was also generalized to non abelian plasmas engineered on D3-D7 systems with D3-branes placed at the tip of a generic singular Calabi-Yau cone. In the zero temperature limit, the resulting backgrounds coincide with those found in [9].333Other solutions employing the smearing technique appear in [10, 11].

In the present paper we extend the above construction to include a finite baryon density (or chemical potential, in the alternative thermodynamical ensemble) for the flavor fields. Working on this problem with holographic techniques is especially interesting, taking also into account that there is no systematic way of dealing with finite baryon density in strongly coupled QCD (lattice QCD suffering from the so-called sign problem).

We provide a novel gravity solution, dual to the above class of flavored plasmas in the planar limit at strong ’t Hooft coupling. While the equations of motion we derive are completely general, the solution can be given in closed analytic form up to second order in (where is the ’t Hooft coupling at the temperature of the plasma) and , where is the baryon density. The gauge theories we focus on become pathological at some UV scale, developing a Landau pole. This is signaled, for example, by a running dilaton (accounting for the breaking of conformal invariance induced by the flavor fields) blowing up at a finite radial value. Correspondingly, the dual gravity solutions are not reliable close to that scale. Keeping small allows both to focus on a regime where the solutions are reliable and to decouple the IR physics - which is the regime we focus on - from the pathological UV behavior.

We also consider the regime of non-large baryon density, , both because it is the relevant regime for the RHIC and LHC experiments and because it allows to derive an analytic solution. Exploring the regime requires a numerical analysis, that we plan to provide in the near future.

The main results and the outline of the paper are as follows. In section 2 we present the action and the ansatz for the D3-D7 setup at finite baryon density. A set of second order differential equations is given in terms of the functions of the radial variable appearing in the ansatz. In section 3 we solve the equations analytically, in a perturbative expansion in and up to second order in both parameters. In section 4 we perform the study of the thermodynamics of the system in the canonical and grand-canonical ensembles, checking the (non-trivial) closure of the various thermodynamic relations. We then explore the effects of the baryon number density on the energy loss of probes through the plasma, in particular on the jet quenching parameter. While the overall effect of flavors is to enhance the jet quenching [3], the effect of finite baryon density depends on the specific choice of comparison scheme of different theories. We finally provide some considerations on certain “optical” properties of the plasma, thinking about the possible gauging of the global . Section 5 contains some concluding remarks. We also provide an appendix with some details on the ten-dimensional action, the equations of motion, the Bianchi identities and their solutions.

The backgrounds we provide correspond to charged black holes in (slightly deformed) , the charge being dual to a finite baryon density. The regime of validity is completely specified and the solution is totally reliable in that regime – there are no uncontrolled approximations. It is the first solution of this kind in the literature and thus it is suitable for the study of a number of physical effects of the baryon density. We hope to explore further the physics of this system in the future.

2 Ansatz and effective Lagrangian

The field theories we focus on are realized on the 4d intersection of “color” D3 and homogeneously smeared “flavor” D7-branes. The D3-branes are placed at the tip of a Calabi-Yau (CY) cone over a Sasaki-Einstein manifold , the latter being a fiber bundle over a four dimensional Kähler-Einstein (KE) base. The ambient spacetime, a product of 4d Minkowski and the CY cone, will be deformed by the backreaction of both kind of branes which respectively source a (self dual) and a RR field. As a result the 10d metric will be in the form of a warped product and there will be a running dilaton. Moreover, the backreaction of the D7-branes will induce a squashing between the KE base of the Sasaki-Einstein manifold and the fibration [9].

Finite temperature is realized by placing a black hole in the center of the background [3]. The D7-branes extend along the radial direction up to the black hole horizon. Their embedding is described by a constant profile, implementing massless flavor fields in the dual gauge theories. In this work we are interested in switching on a chemical potential for the baryon symmetry. The dual picture involves a non-vanishing profile for the temporal component of the worldvolume gauge field on the D7-branes [12]. Through the Chern-Simons coupling, this field can source and form fields.

All in all we will be dealing with a general type IIB action given, in Einstein frame, by

(2.1)

where

(2.2)

is the contribution of the flavor D7-branes. The gravitational constant and D7-brane tension and charge are, in terms of string parameters

(2.3)

The smearing procedure [5, 6] amounts to a replacement

(2.4)

for any form defined on the brane worldvolume.444The smearing of the DBI part of the flavor branes is described at length in [9]. Here is a form orthogonal to the individual location of the D7-branes. For an arbitrary Sasaki-Einstein space , it is proportional to the Kähler form of the Kähler-Einstein 4d basis [9]

(2.5)

For massless flavors, is a constant encoding the density of D7-branes in the relative quotient space with the subspace wrapped by each of the branes

(2.6)

The equations of motion and Bianchi identities that follow from the action (2.1), (2.2) are given in appendix A.

Concerning the metric, we will consider the following ansatz, which includes a family of generalized squashed Sasaki-Einstein manifolds

(2.7)

where the Kähler two-form of the four dimensional base is given in terms of the connection one-form as . The ansatz (2.7) contains two squashing functions and (with dimension of length), whose quotient parameterizes the effect of the flavor backreaction. The dimensionless functions and account for the warping and the blackening of the spacetime, respectively. Thus, in particular, an ansatz with is appropriate for the zero-temperature, uncharged solution. We have used the invariance under diffeomorphisms to choose a convenient holographic radial direction, (with dimension of length), and as we will see, in the IR (UV). Given the smearing procedure (2.4) all the functions in our ansatz depend only on the radial variable.

The finite baryon density is dual to a nontrivial worldvolume gauge field,

(2.8)

A consistent ansatz for the other fields is

(2.9)
(2.10)

In these expressions, is the volume form of , is proportional to the number of colors

(2.11)

is a constant (of dimension length) which we will show to be related to the baryon density, whereas is a function (of dimension length) that describes the effects of the backreaction;555Obviously, . its contribution is dictated by the potential

(2.12)

which is the natural D5 charge sourced on the world-volume of the D7-branes by the gauge field through the last term in (2.2).666Notice that, with the smearing, . In [13] the system without backreacting flavors was studied, and an ansatz for was used that only contained the piece proportional to . However, for our equations of motion to be consistent, we need the presence of the other components; thus, we see that naturally contains the effects of the backreaction of the flavors.

Inserting the whole ansatz into the 10d equations of motion and Bianchi identities one finally arrives at a system of equations which the reader can find in formulas (A.13)–(A.19). It is possible to describe the whole system in terms of an effective one-dimensional action from which the equations of motion can be derived

(2.13)

where denotes the (infinite) integral over the Minkowski coordinates, and

The constraint equation (A.20) is the zero energy condition for the Hamiltonian

(2.15)

Since the gauge field enters only through its derivative it leads to a “constant of motion”. In principle this is a new free parameter which is related to the charge density. However the equations of motion link this to the value of in the ansatz for in (2.9) as we now show. Let us fix this constant of motion as follows

(2.16)

Solving for it gives

(2.17)

Exactly the same expression is obtained from the equation of motion for the form field (see eq. (A.13)). Thus, by enforcing the integration constant as in (2.16) for consistency, we are putting the system partially on shell. On the other hand, this obscures the analysis when it comes to computing the thermodynamical potentials holographically, since it means that the canonical momentum conjugate to was already present in the original Lagrangian. We will comment on this later on.

It is natural to use equation (2.17) to eliminate in favor of and this is usually done in one of two ways: obtaining the equations of motion from (2) and then imposing eq. (2.17), or else, performing a Legendre transformation to the Lagrangian

(2.18)

and then taking the Euler-Lagrange equations from the transformed action. Either way the equations of motion coincide and are given by (A.13)–(A.19) in appendix A.

3 The perturbative solution

In the uncharged case , the following exact solutions for the functions and are readily found [3]: , where is an integration constant of dimension of length. The black hole horizon is at and the extremal limit is reached sending . In terms of a more standard radial coordinate , defined in such a way that with , one gets precisely as for the unflavored black hole. The horizon radius is related to the temperature of the black hole. The whole solution in [3] also depends on the dimensionless combination , which weighs the backreaction of the D7-branes and, in fact, can be read as a flavor-loop counting parameter in the dual field theory.

Now, runs as the dilaton and thus (as a common feature of backreacted D3-D7 setups) it blows up at a finite scale (corresponding to a UV Landau pole in the dual field theory), rendering the supergravity approximation not reliable. Keeping small requires restricting the validity range of our solution up to an arbitrary cutoff , such that

(3.1)

where , , and we have used (2.11). In the uncharged case [3], this condition was used to find an analytic perturbative solution, up to order , for the remaining functions appearing in the ansatz. The related integration constants were fixed requiring regularity at the horizon and matching with the solution [9] at the UV cutoff . The resulting functions thus contained the dimensional (resp. dimensionless) parameters (resp. ). The UV cutoff dependent terms resulted to be of the form of both power-like and logarithmic corrections. Formally sending the arbitrary cutoff scale to infinity, the first kind of corrections drops out, while the second kind can be handled taking into account that the function has a logarithmic running (accounting for the breaking of conformal invariance induced at the quantum level by the massless flavors) such that

(3.2)

This procedure allows to decouple the IR physics from the UV one and to write down a set of solutions containing just and as parameters.

In the present charged case, we are going to follow the very same procedure. Here we have a further parameter to deal with: we will call it , and we will show that it is related to the dimensionless combination of temperature and baryon chemical potential (or charge density, depending on the thermodynamical ensemble). We will then derive an analytical perturbative solution taking both and to be much smaller than one, deforming the finite temperature flavor backreacted solution obtained in [3].

As a first step, let us introduce a dimensional parameter and consider the following redefinitions

(3.3)

The reason behind this choice will be clear in a moment. Inserting these expressions into (A.14)–(A.19) and rewriting the dilaton as , with , one readily arrives at the following system of equations

(3.4)

with

(3.5)
(3.6)
(3.7)

The constraint equation (A.20) reads

(3.8)

The system (3.4)–(3.8) allows for a systematic expansion of all the functions in powers series of and . This is essentially the main effect of the scaling relations (3.3). Once all the functions have been solved for, the worldvolume gauge field can be obtained from the following relation

(3.9)

which is already first order in . From this, we also deduce (as previously announced) that takes the effects of the flavor backreaction into account.

In order to integrate the system (3.4)–(3.8) it is easier to switch to a radial coordinate with an arbitrary parameter of dimension of length. The dimensionless parameter referred to as above is then defined as

(3.10)

where the factor is introduced in order to make it precisely of ref. [12] (see also the equation (4.7) in this paper). We keep the greek symbol, however, in order to stress that our parameter is going to be perturbatively small.

Following the same procedure as in the uncharged case, we are now able to provide analytic solutions to the equations above, in a perturbative expansion in and . We will skip here the intermediate step where the solutions contain cutoff-dependent terms (this can be found in appendix A up to order in eqs. (A.22) to (A.26)) and focus on their effective IR expressions. Introducing the IR parameter and the radial variable (defined again in such a way that the warp factor keeps the standard form), they read

(3.11)
(3.17)
(3.18)

where is an hypergeometric function and is a polylogarithmic function. Notice that enters at order in the equations, hence only the leading contribution in is relevant in the solution.

The above solution must be supplemented with a Jacobian factor for the change of radial coordinate which will show up in the coefficient of

(3.19)

4 Physical properties of the dual plasmas

4.1 Thermodynamics

In the previous section we have derived a solution that, in essence, is a black hole dressed by a set of scalar (dilaton), vector (Maxwell), as well as higher rank tensor fields. Now we are going to extract the thermodynamical properties of the solution, providing, in turn, a first non trivial validity check of the latter by verifying the closure of the standard thermodynamical formulae. As in [3], all quantities are obtained in power series of our perturbative expansion parameters and, therefore, the relevant thermodynamic relations can only be verified up to the relevant order.

To begin with, let us stress that does not coincide with the horizon radius . This radius is defined by and is perturbatively shifted by the baryon density777In fact, there are two radii that solve . This is reminiscent of the case in the Reissner-Nordstrom black hole. The value presented here corresponds to the external radius , i.e. the event horizon. Numerically we have checked that as increases these two radii approach each other. However, a potentially extremal black hole cannot be obtained within the range of validity of our solution, which is perturbative in .

(4.1)

Notice that . Notice moreover that

(4.2)

so that we can trade for , for , and so on, in all of our expressions. In particular, looking at (3.18), we get, to leading order in our expansion

(4.3)

which vanishes at the horizon as required to ensure IR regularity.

The temperature can be computed in the usual way giving

(4.4)

The entropy density is derived from the horizon’s area

(4.5)

As for other thermodynamical variables, this expression, at first order in and , precisely reproduces (for the case of massless flavors) the one found in the probe approximation in [12]. The terms are instead completely new.

Concerning the charge density, a proper value is given by the integration constant in (2.16), as its definition coincides precisely with the electric field displacement. In terms of scaling constants we have, to leading order

(4.6)

and making use of (2.6) as well as (4.4) this may be casted in the form

(4.7)

Using (3.1) with this can be written as

(4.8)

At leading order, and for the case and , we exactly recover formula (A.11) in [12].888Beware that . In the expressions above, is the quark density of the system, related to the baryon density by the number of colors .

4.2 Thermodynamical Potentials

We proceed now to the calculation of the Helmholtz and Gibbs free energies, and respectively.999The difference of these quantities with respect to the forms defined in section 2 should be clear from the context. These can be either directly evaluated starting from the expression for the entropy density and using the standard thermodynamical relations, or they can be deduced holographically. In the latter case they are identified with the (renormalized) on-shell boundary action for the gravity background, evaluated in the corresponding ensemble. Consistency of the solution requires that whatever method is chosen the results are the same.101010In the following we will not report the details of the holographic calculations, but we will just give a sketch of the needed ingredients. Needless to say, we have verified that the results deduced from thermodynamical relations agree with those found from holography.

A quick look at the 1 effective Lagrangian given in (2) reveals that on one hand is a cyclic coordinate and, on the other hand, is a Lagrange multiplier. Amusingly enough, the fact that the equations of motion for and are consistent with one another imposes (2.16), which is nothing but the statement that, up to constant factors, and are canonically conjugate variables. Hence, as it stands, contains both and , hence velocities and momenta. As a consequence the associated action corresponds to neither the canonical nor the grand-canonical ensemble.

4.2.1 Canonical ensemble

The Legendre transformed Lagrangian given in (2.18) is the natural one to describe the system in the canonical ensemble, since it is fully expressed in terms of the baryon density parameter . Therefore, evaluating the associated action on-shell we should obtain the Helmholtz free energy, . As it is well-known, the action has to be supplemented with the standard Gibbons-Hawking term to deal with a well-posed variational problem. Even with this addition, the evaluation presents divergences which we deal with by subtracting the same quantity evaluated on the Euclidean solution at the same temperature but without a horizon and also with no chemical potential. This procedure is explained in appendix B of [3], where we refer the reader for details. We obtain the Helmholtz free energy density

(4.9)

which, consistently, satisfies the thermodynamic relation . To check this relation it is very important to consider the dependence of and on .

The logarithmic running of (see also eq. (3.2)) and the map between the horizon radius and give [3]

(4.10)

Moreover, for one gets by the same token

(4.11)

In the canonical ensemble we must keep the physical (dimensional) charge density invariant. From equation (4.8) we see that the dependence of on at fixed comes from solving as

(4.12)

with a -independent constant. Using (4.11) we obtain