1 Introduction
Abstract

Using a ten dimensional dual string background, we study aspects of the physics of finite temperature large four dimensional gauge theory, focusing on the dynamics of fundamental quarks in the presence of a background magnetic field. At vanishing temperature and magnetic field, the theory has supersymmetry, and the quarks are in hypermultiplet representations. In a previous study, similar techniques were used to show that the quark dynamics exhibit spontaneous chiral symmetry breaking. In the present work we begin by establishing the non–trivial phase structure that results from finite temperature. We observe, for example, that above the critical value of the field that generates a chiral condensate spontaneously, the meson melting transition disappears, leaving only a discrete spectrum of mesons at any temperature. We also compute several thermodynamic properties of the plasma.

arXiv:0709.????



Finite Temperature Large Gauge Theory

with

Quarks in an External Magnetic Field



Tameem Albash, Veselin Filev, Clifford V. Johnson, Arnab Kundu



Department of Physics and Astronomy

University of Southern California

Los Angeles, CA 90089-0484, U.S.A.


talbash, filev, johnson1, akundu [at] usc.edu


1 Introduction

In recent years, the understanding of the dynamics of a variety of finite temperature gauge theories at strong coupling has been much improved by employing several techniques from string theory to capture the physics. The framework is that of holographic [1] gauge/gravity duality, in which the physics of a non–trivial ten dimensional string theory background can be precisely translated into that of the gauge theory for which the rank () of the gauge group is large[2, 3, 4, 5], while the number () of fundamental flavours of quark is small compared to (see ref.[6]). Many aspects of the gauge theory, at strong ’t Hooft coupling , become accessible to computation since the string theory background is in a regime where the necessary string theory computations are classical or semi–classical, with geometries that are weakly curved[2] (characteristic radii in the geometry are set by ).

These studies are not only of considerable interest in their own right, but have potential phenomenological applications, since there are reasons to suspect that they are of relevance to the dynamics of quark matter in extreme environments such as heavy ion collision experiments, where the relevant phase seems to be a quark–gluon plasma. While the string theory duals of QCD are not known, and will be certainly difficult to obtain computational control over (the size of the gauge group there is small, and the number, , of quark flavours is comparable to ) it is expected (and a large and growing literature of evidence seems to support this – see below) that there are certain features of the physics from these accessible models that may persist to the case of QCD, at least when in a strongly coupled plasma phase. Well–studied examples have included various hydrodynamic properties, such as the ratio of shear viscosity to entropy, as well as important phenomenological properties of the interactions between quarks and quark jets with the plasma. Results from these sorts of computations in the string dual language have compared remarkably well with QCD phenomenological results from the heavy ion collision experiments at RHIC (see e.g., refs. [9, 10, 11, 12, 13, 14]), and have proven to be consistent with and supplementary to results from the lattice gauge theory approach.

There are many other phenomena of interest to study in a controllable setting, such as confinement, deconfinement (and the transition between them), the spectrum and dynamics of baryons and mesons, and spontaneous chiral symmetry breaking. These models provide a remarkably clear theoretical laboratory for such physics, as shown for example in some of the early work[7, 8] making use of the understanding of the introduction of fundamental quarks. Some of the results of these types of studies are also likely to be of interest for studies of QCD, while others will help map out the possibilities of what types of physics are available in gauge theories in general, and guide us toward better control of the QCD physics that we may be able to probe using gauge/string duals.

This is the spirit of our current paper111We note that another group will present results in this area in a paper to appear shortly[26].. Here, we uncover many new results for a certain gauge theory at finite temperature and in the presence of a background external magnetic field, building on work done recently[15, 25] on the same theory at zero temperature.

At vanishing temperature and magnetic field, the large gauge theory has supersymmetry, and the quarks are in hypermultiplet representations. Nevertheless, just as for studies of the even more artificial pure gauge theory, the physics at finite temperature — that of a strongly interacting plasma of quarks and gluons in a variety of phases — has a lot to teach us about gauge theory in general, and possibly QCD in particular.

In section two we describe the holographically dual ten–dimensional geometry and the embedding of the probe D7–brane into it. In section three we extract the physics of the probe dynamics, using both analytic and numerical techniques. It is there that we deduce the phase diagram. In section four we present our computations of various thermodynamic properties of the system in various phases, and in sections five and six we present our computations and results for the low–lying parts of the spectra of various types of mesons in the theory. We conclude with a brief discussion in section seven.

2 The String Background

Consider the AdS–Schwarzschild solution given by:

(1)

The dual gauge theory will inherit the time and space coordinates and respectively. Also, in the solution above, is a radial coordinate on the asymptotically AdS geometry and we are using standard polar coordinates on the . The scale determines the gauge theory ’t Hooft coupling according to . For the purpose of our study it will be convenient [7] to perform the following change of variables:

(2)

The expression for the metric now takes the form:

Following ref. [6], we introduce fundamental matter into the gauge theory by placing D7–brane probes into the dual supergravity background. The probe brane is parametrised by the coordinates with the following ansatz for its embedding:

In order to introduce an external magnetic field, we excite a pure gauge –field along the directions [15]:

(3)

where is a real constant. As explained in ref. [15], while this does not change the supergravity background, it has a non–trivial effect on the physics of the probe, which is our focus. To study the effects on the probe, let us consider the general (Abelian) DBI action:

(4)

where is the D7–brane tension, and are the induced metric and induced –field on the D7–branes’ world–volume, is the world–volume gauge field, and here. It was shown in ref. [15] that, for the AdS geometry, we can consistently set the gauge field to zero to leading order in , and the same argument applies to the finite temperature case considered here. The resulting Lagrangian is:

(5)

For large , the Lagrangian asymptotes to:

(6)

which suggests the following asymptotic behavior for the embedding function :

(7)

where the parameters (the asymptotic separation of the D7 and D3–branes) and (the degree of transverse bending of the D7–brane in the plane) are related to the bare quark mass and the fermionic condensate respectively [8] (this calculation is repeated in appendix A). It was shown in ref. [15] that the presence of the external magnetic field spontaneously breaks the chiral symmetry of the dual gauge theory (it generates a non–zero at zero ). However[7], the effect of the finite temperature is to melt the mesons and restore the chiral symmetry at zero bare quark mass. Therefore, we have two competing processes depending on the magnitudes of the magnetic field and the temperature . This suggests an interesting two dimensional phase diagram for the system, which we shall study in detail later.

To proceed, it is convenient to define the following dimensionless parameters:

(8)

This leads to the Lagrangian:

(9)

For small values of , the analysis of the second order, non–linear differential equation for derived from equation (9) follows closely that performed in refs. [7, 16, 17]. The solutions split into two classes: the first class are solutions corresponding to embeddings that wrap a shrinking in the part of the geometry and (when the vanishes) closes at some finite radial distance above the black hole’s horizon, which is located at . These embeddings are referred to as ‘Minkowski’ embeddings. The second class of solutions correspond to embeddings falling into the black hole, since the of the Euclidean section, on which the D7–branes are wrapped, shrinks away there. These embeddings are referred to as ‘black hole’ embeddings. There is also a critical embedding separating the two classes of solutions which has a conical singularity at the horizon, where the wrapped by the D7–brane shrinks to zero size, along with the . If one calculates the free energy of the embeddings, one can show [7, 16, 17] that it is a multi–valued function of the asymptotic separation , which amounts to a first order phase transition of the system (giving a jump in the condensate) for some critical bare quark mass . (For fixed mass, we may instead consider this to be a critical temperature.) We show in this paper that the effect of the magnetic field is to decrease this critical mass, and, at some critical magnitude of the parameter , the critical mass drops to zero. For the phase transition disappears, and only the Minkowski embeddings are stable states in the dual gauge theory, possessing a discrete spectrum of states corresponding to quarks and anti–quarks bound into mesons. Furthermore, at zero bare quark mass, we have a non–zero condensate and the chiral symmetry is spontaneously broken.

3 Properties of the Solution

3.1 Exact Results at Large Mass

It is instructive to first study the properties of the solution for . This approximation holds for finite temperature, weak magnetic field, and large bare quark mass , or, equivalently, finite bare quark mass , low temperature, and weak magnetic field.

In order to analyze the case , let us write for and linearize the equation of motion derived from equation (9), while leaving only the first two leading terms in . The result is:

(10)

Ignoring the terms in equation (10), the general solution takes the form:

(11)

where we have taken . By studying the asymptotic behavior of this solution, we can extract the following:

(12)

By inverting the expression for , we can express in terms of :

(13)

Finally, after going back to dimensionful parameters, we can see that the theory has developed a fermionic condensate:

(14)

The results of the above analysis can be trusted only for finite bare quark mass and sufficiently low temperature and weak magnetic field. As can be expected, the physically interesting properties of the system should be described by the full non–linear equation of motion of the D7–brane. To explore these we need to use numerical techniques.

3.2 Numerical Analysis

We solve the differential equation derived from equation (9) numerically using Mathematica. It is convenient to use infrared initial conditions [17, 18]. For the Minkowski embeddings, based on symmetry arguments, the appropriate initial conditions are:

(15)

For the black hole embeddings, the following initial conditions:

(16)

ensure regularity of the solution at the event horizon. After solving numerically for for fixed value of the parameter , we expand the solution at some numerically large , and, using equation (7), we generate the plot of vs . It is instructive to begin our analysis by revisiting the case with no magnetic field (, familiar from refs.[7, 16, 17]. The corresponding plot for this case is presented in figure 1.

Figure 1: The solid curve starting far left (red) represents solutions falling into the black hole, the dotted (blue) curve represents solutions with shrinking . The vertical dashed line corresponds to the critical value of at which the first order phase transition takes place. The solid black curve dropping sharply from above is the function derived in equation (13), corresponding to the large mass limit.

Also in the figure is a plot of the large mass analytic result of equation (13), shown as the thin black curve in the figure, descending sharply downwards from above; it can be seen that it is indeed a good approximation for . Before we proceed with the more general case of non–zero magnetic field, we review the techniques employed in ref. [17] to determine the critical value of . In figure 2, we have presented the region of the phase transition considerably magnified.

Figure 2: The area below the curve has the interpretation of the free energy of the D7–brane; thus the phase transition pattern obeys the “equal area law”— the area of the shaded regions is equal.

Near the critical value , the condensate is a multi-valued function of , and we have three competing phases. The parameter is known[8] to be proportional to the first derivative of the free energy of the D7–brane, and therefore the area below the curve of the vs plot is proportional to the free energy of the brane. Thus, the phase transition happens where the two shaded regions in figure 2 have equal areas; furthermore, for , the upper–most branch of the curve corresponds to the stable phase, and the lower–most branch of the curve corresponds to a meta–stable phase. For , the lower–most branch of the curve corresponds to the stable phase, and the upper–most branch of the curve corresponds to a metastable phase. At we have a first order phase transition. It should be noted that the intermediate branch of the curve corresponds to an unstable phase.

Now, let us turn on a weak magnetic field. As one can see from figure 3, the effect of the magnetic field is to decrease the magnitude of . In addition, the condensate now becomes negative for sufficiently large and approaches zero from below as . It is also interesting that equation (13) is still a good approximation for .

Figure 3: The effect of the weak magnetic field is to decrease the values of and the condensate. Equation (13) is still a good approximation for .

For sufficiently strong magnetic field, the condensate has only negative values and the critical value of continues to decrease, as is presented in figure 4.

Figure 4: For strong magnetic field the condensate is negative. The value of continues to drop as we increase .

If we further increase the magnitude of the magnetic field, some states start having negative values of , as shown in figure 5. The negative values of do not mean that we have negative bare quark masses; rather, it implies that the D7–brane embeddings have crossed at least once. It was argued in ref. [7] that such embeddings are not consistent with a holographic gauge theory interpretation and are therefore to be considered unphysical. We will adopt this interpretation here, therefore taking as physical only the branch of the vs plots. However, the prescription for determining the value of continues to be valid, as long as the obtained value of is positive. Therefore, we will continue to use it in order to determine the value of for which .

Figure 5: For sufficiently high values of there are states with negative , which are considered non-physical. However the equal area law is still valid as long as .

As one can see in figure 6, the value of that we obtain is . Note also that, for this value of , the Minkowski embedding has a non–zero fermionic condensate , and hence the chiral symmetry is spontaneously broken. For , the stable solutions are purely Minkowski embeddings, and the first order phase transition disappears; therefore, we have only one class of solutions (the blue curve) that exhibit spontaneous chiral symmetry breaking at zero bare quark mass. Some black hole embeddings remain meta–stable, but eventually all black hole embeddings become unstable for large enough . This is confirmed by our study of the meson spectrum, which we present in later sections of the paper.

Figure 6: For the critical parameter vanishes. There are two states with equal energies, one of them has non-vanishing condensate and therefore spontaneously breaks the chiral symmetry.

The above results can be summarized in a single two dimensional phase diagram, which we present in figure 7.

Figure 7: The curve separates the two phases corresponding to discrete meson spectrum (light mesons) and continuous meson spectrum (melted mesons).

The curve separates the two phases corresponding to a discrete meson spectrum (light mesons) and a continuous meson spectrum (melted mesons) respectively. The crossing of the curve is associated with the first order phase transition corresponding to the melting of the mesons. If we cross the curve along the vertical axis, we have the phase transition described in refs. [7, 16, 17]. Crossing the curve along the horizontal axis corresponds to a transition from unbroken to spontaneously broken chiral symmetry[15], meaning the parameter jumps from zero to , resulting in non–zero fermionic condensate of the ground state. It is interesting to explore the dependence of the fermionic condensate at zero bare quark mass on the magnetic field. From dimensional analysis it follows that:

(17)
Figure 8: The solid curves are the numerically extracted dependence , while the dashed curve represents the expected large behavior . The solid curve segments at the bottom left and to the upper right (blue) are the stable states. The straight segment and the arc that joins it (lower right, red) and red are the unstable states. The rest (cyan) are meta–stable states.

In the limit, we should recover the result from ref. [15]: , which implies that for . The plot of the numerically extracted dependence is presented in figure 8; for , very fast approaches the curve . This suggests that the value of the chiral symmetry breaking parameter depends mainly on the magnitude of the magnetic field , and only weakly on the temperature .

4 Thermodynamics

Having understood the phase structure of the system, we now turn to the extraction of various of its important thermodynamic quantities.

4.1 The Free Energy

Looking at our system from a thermodynamic point of view, we must specify the potential characterizing our ensemble. We are fixing the temperature and the magnetic field, and hence the appropriate thermodynamic potential density is:

(18)

where is the magnetization density and is the entropy density of the system. Following ref. [19], we relate the on–shell D7–brane action to the potential density  via:

(19)

where (here, ):

(20)

In principle, on the right hand side of equation (19), there should be terms proportional to , which subtract the energy of the magnetic field alone; however, as we comment below, the regularization of is determined up to a boundary term of the form . Therefore, we can omit this term in the definition of . The boundary action contains counterterms designed[20] to cancel the divergent terms coming from the integral in equation (20) in the limit of . A crucial observation is that the finite temperature does not introduce new divergences, and we have the usual quartic divergence from the spatial volume of the asymptotically AdS spacetime [21]. The presence of the non–zero external magnetic field introduces a new logarithmic divergence, which can be cancelled by introducing the following counterterm:

(21)

where is the metric of the 4–dimensional surface at . Note that in our case:

(22)

which gives us the freedom to add finite terms of the form at no cost to the regularized action. This makes the computation of some physical quantities scheme dependent. We will discuss this further in subsequent sections. The final form of in equation (20) is:

(23)

It is instructive to evaluate the integral in equation (20) for the embedding at zero temperature. Going back to dimensionful coordinates we obtain:

(24)

The first two terms are removed by the counter terms from , and we are left with:

(25)

This result can be used to evaluate the magnetization density of the Yang–Mills plasma at zero temperature and zero bare quark mass. Let us proceed by writing down a more general expression for the free energy of the system. After adding the regulating terms from , we obtain that our free energy is a function of :

(26)

where is defined via:

In order to verify the consistency of our analysis with our numerical results, we derive an analytic expression for the free energy that is valid for . To do this we use that for large , the condensate is given by equation (13), which we repeat here:

(28)

as well as the relation . We then have:

(29)

where the function can be obtained by evaluating the expression for from equation (4.1) in the approximation . Note that this suggests ignoring the term , which is of order . Since the leading behavior of at large is , this means that the results obtained by setting can be trusted to the order of , and therefore we can deduce the function , corresponding to the zeroth order term. Another observation from earlier in this paper is that the leading behavior of the condensate is dominated by the magnetic field and therefore we can further simplify equation (4.1):

(30)

Comparing to equation (29), we obtain:

(31)

and our final expression for , valid for :

(32)

4.2 The Entropy

Our next goal is to calculate the entropy density of the system. Using our expressions for the free energy we can write:

(33)

It is useful to calculate the entropy density at zero bare quark mass and zero fermionic condensate. To do this, we need to calculate the free energy density by evaluating the integral in equation (4.1) for . The expression that we get for is:

(34)

The corresponding expression for the entropy density is:

(35)

One can see that the entropy density is positive and goes to zero as . Our next goal is to solve for the entropy density at finite for fixed . To do so, we have to integrate numerically equation (33) and generate a plot of versus . However, for we can derive an analytic expression for the entropy. After substituting the expression from equation (32) for into equation (33) we obtain:

(36)

or if we go back to dimensionful parameters:

(37)

One can see that if we send , while keeping fixed we get the behavior described in ref. [19], and therefore the (approximate; ) conformal behaviour is restored in this limit. In figure 9, we present a plot of versus for . The solid smooth black curve corresponds to equation (36). For this is positive and always a decreasing function of . Hence, the entropy density at fixed bare quark mass , given by , is also a decreasing function of and therefore an increasing function of the temperature, except near the phase transition (the previously described crossover from black hole to Minkowski embeddings) where an unstable phase appears that is characterized by a negative heat capacity.

Figure 9: A plot of versus for . The thin (sharply descending and extending to the right) black curve corresponds to that large mass result of equation (36).

4.3 The Magnetization

Let us consider equation (25) for the free energy density at zero temperature and zero bare quark mass. The corresponding magnetization density is given by:

(38)

Note that this result is scheme dependent, because of the freedom to add terms of the form to the boundary action that we discussed earlier. However, the value of the relative magnetization is given by:

(39)

is scheme independent and is the quantity of interest in the section. In equation (39), we have defined as a dimensionless parameter characterizing the relative magnetization. Details of how the derivative is taken are discussed in appendix B. The expression for follows directly from equation (4.1):

(40)

For the large region we use the asymptotic expression for from equation (32) and obtain the following analytic result for :

(41)

We evaluate the above integral numerically and generate a plot of versus . A plot of the dimensionless relative magnetization versus for is presented in figure 10. The black curve corresponding to equation (41) shows good agreement with the asymptotic behavior at large . It is interesting to verify the equilibrium condition . Note that since does not depend on the temperature, the value of this derivative is a scheme independent quantity. From equations (39) and equation (41), one can obtain:

(42)

which is valid for large and weak magnetic field . Note that the magnetization seems to increase with the temperature. Presumably this means that the temperature increases the “ionization” of the Yang–Mills plasma of mesons even before the phase transition occurs.

Figure 10: A plot of the dimensionless relative magnetization versus for . The thin black curve (starting with a steep descent) corresponds to the large mass result of equation (41).

4.4 The Speed of Sound

It is interesting to investigate the effect of the magnetic field on the speed of sound in the Yang–Mills plasma. Following ref. [19], we use the following definition to thermodynamically determine the speed:

(43)

where is the density of the heat capacity at constant volume. To compute the contribution coming from the fundamental flavors in the presence of an external magnetic field, we work perturbatively in small . First, let us recall the adjoint contribution to entropy and specific heat [19]:

(44)

To proceed, let us rewrite the entropy density of the fundamental flavours in the following form:

(45)

where . The term is simply the contribution from the conformal theory; the deviation from it is related to the conformal symmetry being broken by introducing the fundamental flavors and the external magnetic field. This breaking is manifest by non–vanishing and in equation (45) respectively. Recalling the relation between the energy density and the free energy density , and, using equation (45), we find that:

(46)

Using the definition in equation (43), together with the above results and expanding up to first order in , we get:

(47)

The second and third term in equation (47) represent the deviation from the conformal value of by the presence of the fundamental flavors and the external magnetic field. For convenience let us define . It is possible to obtain an analytic expression for in the limit of large bare quark mass and small magnetic field (). Using our previous analytic expressions, we get:

(48)

It is important to note that equation (48) is valid only up to first order in . To proceed beyond the large bare quark mass and small magnetic field limit, we study numerically the velocity deviation, which is summarised in figure 11.

Figure 11: The deviation of the speed of sound from the conformal value in units of in the presence of fundamental matter and an external magnetic field. The curves coming in from the left (red) correspond to black hole embeddings, and the curves coming in from the right (blue) correspond to Minkowski embeddings. The vertical dashed line represents the phase transition point, and the flatter dashed curves (green) correspond to the approximate analytic expression given in (48). We do not include the curve of the analytic result in 11 since the approximate formula is not valid for high magnetic fields.

We observe from figure 11 that, for small magnetic field, the deviation is similar to the zero magnetic field case; approaches zero (corresponding to restoration of the conformal symmetry) from below in both the and limits. However in presence of large magnetic field (see figure 11), we see that , and the conformal value is never attained.

5 Meson Spectrum

In this section, we calculate the meson spectrum of the gauge theory. The mesons we are considering are formed from quark–antiquark pairs, so the relevant objects to consider are 7–7 strings. In our supergravity description, these strings are described by fluctuations (to second order in ) of the probe branes’ action about the classical embeddings we found in the previous sections[22]. Studying the meson spectrum serves two purposes. First, tachyons in the meson spectrum from fluctuations of the classical embeddings indicate the instability of the embedding. Second, a massless meson satisfying a Gell-Mann-Oakes-Renner (GMOR) relation will confirm that spontaneous chiral symmetry breaking has occurred. As a reminder, in ref. [22], the exact meson spectrum for the AdS background was found to be given by:

(49)

where labels the order of the spherical harmonic expansion, and is a positive integer that represents the order of the mode. The relevant pieces of the action to second order in are:

(50)
(51)