Quark mass and condensate in HQCD
We extend the Sakai-Sugimoto holographic model of QCD (HQCD) by including the scalar bi-fundamental “tachyon” field in the 8-brane-anti-8-brane probe theory. We show that this field is responsible both for the spontaneous breaking of the chiral symmetry, and for the generation of (current algebra) quark masses, from the point of view of the bulk theory. As a by-product we show how this leads to the Gell-Mann-Oakes-Renner relation for the pion mass.
The closest model so far to a holographic description of large QCD is the Sakai-Sugimoto model . The gauge sector is described, at low energy, by the near-horizon limit of D4-branes wrapped on a Scherk-Schwarz circle . This description is valid at energies well below the Kaluza-Klein scale of the circle. The quark sector is incorporated by including D8-branes and anti-D8-branes transverse to the circle. The strings that stretch between the original D4-branes and the D8-branes (anti-D8-branes) describe right-handed (left-handed) chiral fermions, which transform in the fundamental representation of both the color group and the () flavor group. It is assumed that , so that the D8-branes can be treated as probes in the D4-brane background, and one can ignore their backreaction. In QCD this corresponds to the quenched approximation. The Sakai-Sugimoto model shares many features with other holographic models of gauge theory with matter, however the novel feature of this model is the geometrical realization of spontaneous flavor chiral symmetry breaking. The 8-branes and anti-8-branes are separated along the circle asymptotically in the radial coordinate, but are connected at some minimal radial position (figure 1). The former corresponds in QCD to the UV flavor symmetry being , and the latter corresponds to the IR symmetry being the diagonal subgroup .
In spite of its success, there are several open questions about the model, some of which are related to very basic notions of gauge dynamics. The first is the incorporation of a QCD, or “current algebra” quark mass. The quarks are massless in this model since the 8-branes necessarily intersect the 4-branes. This is also manifested in the fact that the modes identified with the pions are massless. It is well known that pions obey the Gell-Mann-Oakes-Renner relation 
Therefore in a non-trivial quark condensate massless pions imply massless quarks. In the original construction of  the 8-branes and anti-8-branes were located at antipodal points on the circle, and they connected at the minimal radial position of the background . This was extended in ,, to a family of configurations parameterized by the asymptotic separation , or equivalently by the minimal radial position of the 8-branes (see figure 1(b)). In these configurations there is a natural mass parameter associated with the “length” of a string stretched from the minimal radial position of the background to the minimal radial position of the 8-branes . However, it is easy to check that the pions remain massless, so this parameter cannot be identified with the current algebra mass. It is instead related to the “constitutent quark mass”. Indeed in  for the model of , and in  for analogous non-critical models, it was found that the masses of the vector mesons are linearly related to this “length of the string” mass parameter. Moreover, a model of the decay process of spinning stringy mesons  supports the interpretation of the “length” as the constituent mass.
A related question has to do with the quark condensate itself: how does one compute it in the holographic description?111In  both the current algebra mass of the quarks and the condensate can be read from the profile of the flavor branes, and the GOR relation of (1) is obeyed. However that model suffers from the drawback that it does not incorporate chiral flavor symmetry. The answers to both questions are related to each other. The quark mass term in QCD is
where is an matrix, is a fundamental of , and is a fundamental of . Both the mass and the quark bi-linear should therefore be identified with a bi-fundamental field in the bulk.222In  the quark mass and condensate were identified with the scalar field corresponding to the 8-brane-anti-8-brane separation. We believe this is incorrect, since this field transforms in the adjoint, rather than the bi-fundamental, representation of the s. In this model the required bi-fundamental field comes from the D8- strings.333The same field appears in the holographic description of the resolution of the puzzle . The dual operator is therefore non-local in the coordinate transverse to the 8-branes. According to the usual holographic dictionary the normalizable mode of this field should correspond to the expectation value of the quark bi-linear, i.e. to the quark condensate, and the non-normalizable mode should correspond to the quark mass.
The bi-fundamental field was not included in the analysis of , since, as was argued there, it is very massive. The 8-brane-anti-8-brane separation was assumed to be much greater than the string length, which in flat space would make this field massive. However the proper distance between the 8-branes and anti-8-branes in the curved background of this model depends on the radial coordinate , and decreases as decreases. The mass of the bi-fundamental field therefore depends on as well. For the U-shaped configuration found in  this field remains massive for all . While the proper distance decreases as decreases, the relative angle between the 8-branes and anti-8-branes increases, so that it never becomes tachyonic. The situation changes in the case of the non-compact background considered in . In that case , so there are two possible configurations: a connected U-shaped 8-brane similar to the one of , and a disconnected parallel 8-brane-anti-8-brane configuration. In the parallel configuration the proper distance between the 8-branes and anti-8-branes goes to zero at , and therefore the bi-fundamental field becomes tachyonic in a finite range of near the origin. This represents a (radially) localized tachyonic instability, and one expects the true vacuum to be the U-shaped configuration444A similar effects occur in the “hairpin brane” of , and in the meta-stable supersymmetry breaking brane configurations of .. The condensation of this localized tachyon can be seen as a Higgs-like effect which breaks the chiral symmetry to the diagonal group .
In either case, the remaining non-tachyonic mode of the bi-fundamental field, which we will continue to call the “tachyon” , is crucial for describing the quark mass and condensate.
In this paper we incorporate the tachyon into the 8-brane action using a proposal of Garousi for the brane-antibrane effective action , which extends Sen’s original proposal for the non-BPS D-branes . We show that the coupled equations of motion for the tachyon and the 8-brane-anti-8-brane separation admit a solution which describes a U-shaped configuration. In our solution the tachyon has a non-trivial profile, which for large is a linear combination of a normalizable mode and a non-normalizable mode. We relate the coefficient of the former to the quark condensate , and the coefficient of the latter to the quark mass . We also show that the pions, which are part of the meson spectrum, acquire a mass that satisfies the GOR relation (1).555A different approach to the pion mass in this model was discussed in . For our solution describes the same configuration as [1, 11], but it also includes the effect of the (normalizable mode of the) massive bi-fundamental field. At large the solutions are the same, but the precise shape of the 8-brane at finite changes.
For simplicity we will consider the non-compact case dual to the NJL model , namely the near-horizon background of extremal D4-branes. The metric in this case does not contain the “thermal factor” . The behavior near the boundary at will be similar to the compact case since , and therefore our results for the quark mass and condensate, which are determined by the behavior of near the boundary, will be the same. This is also reasonable from the field theory point of view. While the gauge sector of this model is very different from QCD, and Kaluza-Klein states do not decouple, the flavor sector, which is where chiral symmetry breaking and quark masses are seen, is the same. We will also deal only with the one flavor case , for which the 8-brane theory is Abelian. Note that in this case the would-be broken symmetry is the anomalous . At large , however, the anomaly, and with it the mass of the would-be Goldstone boson , is suppressed [17, 18] (see however  for a discussion of how it is suppressed in this model). The GOR relation (1) therefore holds also for the . 666There is an alternative large extension of one-flavor QCD, in which the fermions transform in the anti-symmetric representation of the gauge group . In that model the anomaly is not suppressed, and the GOR relation is not expected to hold for the .
A holographic dual description of the chiral condensate and quark mass in QCD has been discussed previously in the context of the “bottom-up” AdS/QCD model [20, 21], which is essentially a five-dimensional Yang-Mills theory in with a bi-fundamental tachyonic scalar field. This was later generalized to a tachyonic DBI + CS theory in .
The outcome of the present paper is a holographic picture where
The spontaneous breaking of flavor chiral symmetry emerges from a “Higgs mechanism” with an order parameter which is the expectation value of the bi-fundamental tachyon field.
The current algebra mass of the quarks is associated with a non-normalizable mode of the tachyon. The quark anti-quark condensate can be identified with a normalizable mode of the tachyon.
The pions of the model obey the GOR relation.
The paper is organized as follows. In section 2 we review the proposal for the D- effective action, and apply it to the D- system in the near-horizon extremal 4-brane background. In section 3 we study the asymptotic forms of the solutions for and , both at large and near the point where the branes and anti-branes connect. We extract the quark mass and condensate from the behavior of at large . In section 4 we present numerical solutions which interpolate between the two asymptotic solutions, and compare with the solution without the tachyon of . In section 5 we begin to analyze the meson spectrum in the tachyon background. This includes both the fluctuations of the scalar fields and , as well as the worldvolume gauge fields on the 8-branes and anti-8-branes. In particular we show that the mass of the pions satisifies the GOR relation.
2 The D- theory
A proposal for the effective action of a parallel -brane-anti--brane system in curved spacetime was given by Garousi in . Denoting by and the adjoint (position) scalar fields and gauge fields on the branes () and anti-branes (), and by the complex bi-fundamental scalar field, the action is given by
where is the brane-antibrane separation, and
We use for the worldvolume directions, and for the transvese directions. The covariant derivative of the bi-fundamental scalar is given by , and is the scalar field potential.
This action was obtained by generalizing Sen’s action for a non-BPS 9-brane in Type IIA string theory  as follows. First, the tachyon kinetic term is added under the square root . Second, the action is extended to two unstable 9-branes by the familiar symmetric trace prescription for the non-Abelian DBI action. Third, the action is transformed to a 9-brane-anti-9-brane action in Type IIB string theory by projecting with . Finally, the general -brane-anti--brane action is obtained by T-duality. To separate the branes and antibranes we turn on a Wilson line in the 9-brane-anti-9-brane model, which fixes the dependence on . As a check, note that for this reduces to Sen’s action for a coinciding brane and antibrane .
The tachyon potential for the brane-antibrane pair is not known precisely even in flat space. Boundary superstring field theory gives a potential 
This reproduces, for example, the S-brane solution using the tachyon effective theory . In both proposals (and there may be others) the true vacuum is at , but the details are different. We will work with the inverse cosh potential (7) shown in figure 2.
Let us apply this proposal to the D8- system in the non-compact extremal D4-brane background. The background is defined by
where , and . We assume that the 8-brane and anti-8-brane are positioned symmetrically at and , respectively, and that the configuration depends only on the radial coordinate . It will also be convenient to work in the unitary gauge in which is real. Suppressing the gauge fields for now, the action for the D8- pair in this background becomes
and where we have defined and .
Note that the proper distance between the 8-brane and anti-8-brane is
so even if we keep the brane and antibrane well separated in coordinate distance, the proper distance will decrease below the string scale for small enough , and the field will be tachyonic in that region. One can see this directly by expanding the action for small , which gives (after properly normalizing to get a canonical kinetic term)
We recognize the first term as the zero-point energy of the open superstring in the NS sector in flat space, and the second term as the contribution of the proper length of the open string. This result is most likely not precise. First, the flat space result for the zero-point energy probably changes in this background. We do not know how to compute it, since this is an RR background. Second, the straight string stretched between the 8-brane and anti-8-brane is not the minimal length (and mass) string, and it actually prefers to curve down in . However we believe that the qualitative result is still correct, namely that below some critical . In other words the field has a localized tachyonic mode in a small region near . This is similar to the tachyon which appears at the intersection of branes which meet at a small angle. We therefore expect the ground state to correspond to the connected 8-brane configuration.
2.1 The compact case
In the Sakai-Sugimoto model is compact and the near-horizon metric is given by
Strictly speaking, the 8-brane and anti-8-brane cannot be treated as separate entities in this background, since the circle shrinks to zero size at . In this case the brane and anti-brane are necessarily connected, and the D8- action should be viewed as a large- effective theory for the worldvolume fields on the two sides of the 8-brane, together with the massive scalar field coming from the open string stretched between the two sides. At large the compact background is essentially identical to the non-compact one, so the results related to the mass and condensate will be the same. The precise profiles of the fields and at finite will be different.
3 Asymptotic solutions
The equations of motion that follow from (10) are given by
where was defined in (11), is the inverse cosh potential, and we have set .
The tachyon equation (17) has a trivial solution . In this case the solution to the equation (16) is , corresponding to the parallel D8- configuration.777Equation (16) reduces in this case to the same equation one gets from the single 8-brane action without the tachyon . There are two solutions in that case corresponding to a straight 8-brane and a U-shaped 8-brane. However the action in our case is doubled since it includes both an 8-brane and an anti-8-brane. Consequently there are four possible solutions with , corresponding to either brane or antibrane being straight or U-shaped. We are interested only in solutions with two asymptotic boundaries, one for the 8-brane and one for the anti-8-brane (or equivalently we require and to be single-valued). With that leaves only the straight and parallel D8- solution. This configuration is unstable due to the localized tachyon mode near . The stable solution must involve a non-trivial tachyon condensate , which, as we shall see below, corresponds to a single U-shaped configuration.
We expect the 8-brane and anti-8-brane to connect roughly at the radial position below which the bi-fundamental field is tachyonic. Let us first expand the fields near this point:
where we assume that . To leading order, the equation (16) gives
This implies, in particular, that . In addition, the absence of sources at implies that , so that and the configuration is smooth. The equation (17) is then also satisfied to leading order. The leading behavior near is then
This is in accord with the interpretation of the non-trivial solution as the chiral-symmetry-breaking U-shaped configuration. The brane-antibrane separation vanishes at , and the tachyon diverges, i.e. goes to its true vacuum in the potential (7).
To compute the gauge theory quantities, in this case the quark mass and condensate, we should look at the behavior of the solution at large . This corresponds to the UV limit of the gauge theory. Strictly speaking, the UV limit is not well-defined in this model, since it is really a five-dimensional gauge theory. We will therefore always be considering a UV cutoff . In this regime the field is very massive, so we can consider small fluctuations away from the trivial solution:
where and . In this approximation the action is quadratic
The asymptotic solutions for are given by
These solutions are only valid in a regime of for which the fluctuations are small. We therefore have to assume that , whereas and can be taken to be in the cutoff.
3.1 Quark mass and condensate
The growing and decaying exponentials correspond to the non-normalizable and normalizable solutions for , respectively. We would therefore like to identify the coefficients and with the quark mass and quark condensate , respectively. Let us verify this explicitly. In QCD (at zero temperature) the quark condensate is given by the variation of the energy density with respect to the quark mass
Let us assume that is given by the (dimensionless) parameter
where is some fixed mass scale. To evaluate (27) in the holographic dual we must vary the asymptotic (Euclidean) 8-brane action (24) with respect to the parameter of the solution. The general variation is
Using the equations of motion this reduces to
Focussing on the variation with respect to the tachyon we find
Only the upper limit contributes, since although both and diverge in the lower limit, the potential much faster. Using the large asymptotic form of the solution (26) we find for a variation with respect to :
where we have imposed the cutoff . Since is identified with the quark mass, we find that the quark condensate is related to as
4 Numerical solutions
The asymptotic solutions near and at large must connect in the full solution to the equations of motion (16) and (17). In this section we present a numerical analysis of these equations. For convenience we define the dimensionless quantities (recall that )
In terms of these the D8- action (10) becomes
and the equations of motion become
The range for will be approximated numerically by the range .
The solution is fixed by imposing boundary conditions for , , and their derivatives, either at infinity (UV) or at (IR), which corresponds roughly to . Let’s look at UV boundary conditions first. Guided by the UV asymptotic form of the solution (25) and (26), we impose
Figure 3(a) shows the resulting numerical solution for the shape of the 8-brane, and figure 3(b) shows the tachyon profile for this solution. The tachyon increases as decreases, and blows up, i.e. attains its vacuum value, where the 8-brane and anti-8-brane connect.
Now let’s look at IR boundary conditions. Guided by the IR asymptotics (21) we impose the numerical boundary conditions
The solution near the connection point at is shown in figure 4.
The behavior is qualitatively the same as with the UV boundary conditions. When we look at larger , however, the qualitative behavior changes (figure 5). This is due to the sensitivity of the numerical solution to the IR boundary conditions (40). The exact IR boundary values of and are infinite. As we increase the numerical IR boundary values of and the region where the behavior changes moves to larger and larger .
4.1 A comparison with the AHJK solution
Using the same UV boundary conditions we arrive at the numerical solution presented in figure 6, where we also present our solution for comparison. The two configurations are very close in this regime, which is understandable since the correction due to the very massive field is small.
We would like to argue that our solution with the non-trivial tachyon profile is, in some sense, a better approximation to the exact string theory solution. As evidence for this we will show that the free energy of our solution is smaller than that of the solution of , which we will refer to as the AHJK solution.888Recall also that the U-shaped solution of  has a lower free energy than that of the parallel configuration. The free energy is given by the Euclidean action of the solution, which for our solution is
The free energy for the AHJK solution is given by
By splitting the integration over to a UV part and an IR part, we can estimate the free energy of our tachyonic solution and the AHJK solution. In the IR region, the free energy of the tachyonic solution is strongly suppressed by the exponentially vanishing factor , which comes from the tachyon potential. So is obviously much smaller than the IR part of the AHJK free energy . On the other hand, in the UV region, we shall compare and numerically. From figure 6(b), and using the numerical solutions of , we calculate999 The lower bounds of the integration intervals depend on the capacity of our computer.
that is to say, . Combining the results in the IR and UV regions, we see that
so our tachyonic solution appears to be more favorable than the AHJK solution.
5 The meson spectrum
We now turn to the analysis of the spectrum in the tachyon U-shaped background. The fluctuations of and the gauge fields on the 8-brane and anti-8-brane correspond to various mesons, including scalars, pseudo-scalars, vectors and axial-vectors. We are interested mainly in the lowest pseudo-scalar modes (the pions). In particular, we would like to see how they acquire mass when the quarks are massive. For completeness we also set up the eigenvalue problems for the other mesons, but we leave the (numerical) analysis for future work.
5.1 Scalar fields
Let’s start with the scalar fields and . We expand around the classical solution
where and are real scalar fields101010In general there is also a pseudo-scalar fluctuation of the phase of the tachyon , but we are working in unitary gauge where .. Expanding the 8-brane-anti-8-brane action to quadratic order gives
where the coefficients are given by
The four-dimensional mass matrix will get contributions from all the quadratic terms in .
5.2 Gauge fields
Now consider the gauge fields and . We will use the symmetric and anti-symmetric combinations
and consider only the and components. The former will give rise to vector () and axial-vector () mesons, and the latter to scalar and pseudo-scalar mesons in four dimensions. Let us also further fix the gauge by setting
Expanding the action to quadratic order in the gauge fields then gives
5.2.1 The sector
The action in the symmetric (vector) sector is given by
This sector is similar to the gauge field in the single 8-brane case in . We expand the gauge field in radial modes
that satsify the eigenvalue equation
and the normalization condition
The four-dimensional action in this sector is then
The zero mode , with , is special. The eigenvalue equation (54) gives
In the UV asymptotic region this becomes
so is UV-normalizable. On the other hand, in the IR asymptotic region we get
The exponential divergence of the integrand as implies that is non-normalizable in the IR. We therefore have to exclude this mode from the spectrum. The spectrum of vectors is therefore purely massive. Note that the exponential divergence is due to the tachyon.
5.2.2 The sector
The action in the anti-symmetric sec