Holography Inspired Stringy Hadrons
Holography inspired stringy hadrons (HISH) is a set of models that describe hadrons: mesons, baryons and glueballs as strings in flat four dimensional space time. The models are based on a “map” from stringy hadrons of holographic confining backgrounds. In this note we review the “derivation” of the models. We start with a brief reminder of the passage from the string theory to certain flavored confining holographic models. We then describe the string configurations in holographic backgrounds that correspond to a Wilson line,a meson,a baryon and a glueball. The key ingredients of the four dimensional picture of hadrons are the “string endpoint mass” and the “baryonic string vertex”. We determine the classical trajectories of the HISH. We review the current understanding of the quantization of the hadronic strings. We end with a summary of the comparison of the outcome of the HISH models with the PDG data about mesons and baryons. We extract the values of the tension, masses and intercepts from best fits, write down certain predictions for higher excited hadrons and present attempts to identify glueballs.
- 1 Introduction
- 2 From to confining string backgrounds
3 Hadrons as strings in holographic background
- 3.1 The holographic Wilson line
- 3.2 The stringy duals of mesons
- 3.3 Holographic stringy baryons
- 3.4 The glueball as a rotating closed string
- 4 Holography inspired hadronic strings
5 On the quantum stringy hadrons
- 5.1 The quantum rotating closed string in holographic background
- 5.2 On the quantization of the string in non-critical dimensions
- 5.3 The Casimir energy of a critical static string with massive endpoints
- 5.4 On the quantization of the rotating string with massive endpoints in four dimensions
6 Phenomenology: Comparison between the stringy models and experimental data
- 6.1 Fitting models and procedure
- 6.2 Review of meson trajectory fits
- 6.3 Review of baryon trajectory fits
- 6.4 Strings and the search for glueballs
- 7 Summary and future directions
The stringy description of hadrons has been thoroughly investigated since the sixties of the last century. In this paper I review the research work we have been doing in the last few years on a renewed stringy description of hadrons. This naturally raises the question of what are the reasons to go back to “square one” and revisit this question? Here are several reasons for doing it.
(i) Up to date, certain properties, like the hadronic spectrum, the decay width of hadron and their scattering cross section, are hard to derive from QCD and relatively easy from a stringy picture.
(ii) Holography, or gauge/string duality, provides a bridge between the underlying theory of QCD (in certain limits) and a bosonic string model of hadrons .
(iii) To establish a framework that describes the three types of hadrons mesons, baryons and glueballs in terms of the same building blocks, namely strings.
(iv) The passage from the holographic string regime to strings in reality is still a tremendous challenge.
(v) Up to date we lack a full exact procedure of quantizing a rotating string with massive endpoints (which will see are mandatory for the stringy hadrons).
(vi) There is a wide range of heavy mesonic and baryonic resonances that have been discovered in recent years. Thus the challenge is to develop a framework that can accommodate hadrons with any quark content light, medium and heavy.
The holographic duality is an equivalence between certain bulk string theories and boundary field theories. Practically most of the applications of holography are based on relating bulk fields (not strings) and operators on the dual boundary field theory. This is based on the usual limit of with which we go, for instance, from a closed string theory to a theory of gravity. However, to describe realistic hadrons it seems that we need strings since after all in nature the string tension which is inversely proportional to is not very large. In holography this relates to the fact that one needs to describe nature is of order one and not very large one.
The main theme of this review paper is that there is a wide sector of hadronic physical observables which cannot be faithfully described by bulk fields but rather require dual stringy phenomena. It is well known that this is the case for a Wilson, a ’t Hooft and a Polyakov lines ( for a review see for instance ). We argue here that in fact also the spectra, decays and other properties of hadrons: mesons, baryons and glueballs can be recast only by holographic stringy hadrons and not by fields that reside in the bulk or on flavor branes. The major argument against describing the hadron spectra in terms of fluctuations of fields like bulk fields or modes on probe flavor branes is that they generically do not admit properly the Regge behavior of the spectra. For as a function of we get from flavor branes only , mesons and there is a big gap of order ( or certain fractional power of depending on the model) in comparison to high mesons described in terms of strings. Moreover the attempts to get the observed linearity between and the excitation number is problematic, whereas for strings it is an obvious property.
The main ideas of this project is (i) To analyze string configurations in holographic string models that correspond to hadrons, (ii) to bypass the usual transition from the holographic regime of large and large to the real world via a and expansion and state a model of stringy hadrons in flat four dimensions that is inspired by the corresponding holographic strings. (iii) To confront the outcome of the models with the experimental data in , .
Confining holographic models are characterized by a “wall” that truncates in one way or another the range of the radial direction (see figure 2). A common feature to all the holographic stringy hadrons is that there is a segment of the string that stretches along a constant radial coordinate in the vicinity of the “wall”. For the stringy glueball it is the whole folded closed string that rotates there and for the open string it is part of the string, the horizontal segment, that connects with vertical segments either to the boundary for a Wilson line or to flavor branes for the meson and for the baryon. This fact that the classical solutions of the flatly rotating strings reside at fixed radial direction is behind the map to rotating strings in flat four dimensional space-time. A key ingredient of the map is the , the “ string endpoint mass” that provides in the four flat space-time description the dual of the vertical string segments. It is important to note (i) This mass parameter is neither the QCD mass parameter nor that of the constituent quark mass. (ii) As will be seen below the parameter is not an exact map of a vertical segment but rather only an approximation that is more accurate the longer the horizontal string is.
As we have mentioned above the stringy picture of meson has been thoroughly investigated in the past and we will not cite here this huge body of papers. It turns out that also in recent years there were several attempts to describe hadrons in terms of strings. Papers on the subject that have certain overlap with our approach are for instance ,,,. Another approach to the stringy nature of QCD is the approach of low-energy effective theory on long strings. This approach is different but shares certain features with the approach presented in this paper. A recent review of the subject can be found in .
The alternative description of hadrons in terms of fields in the bulk or on flavor branes is not discussed in this paper. For a review paper about this approach and reference therein see  The paper is organized in the following manner. After this section of an introduction there is a review section that describes the passage from the string background to that of various confining backgrounds. We describe in some details the prototype model of Sakai and Sugimoto and its generalization. Section is devoted to hadrons as strings in a holographic background. In this section we separately describe in the holographic Wilson line, in the stringy duals of mesons and in the glueball as a rotating folded closed string. We then describe in section the HISH model. We present the map between the holographic strings and strings in flat space-time in . We classically solve the system of an open string with massive endpoints and we determine its energy and angular momentum . We then in discuss the stringy baryon of HISH and in particular the stability of Y shape string configurations. Next in section we present our current understanding of the quantization of the HISH model. In we review the attempt to quantize the closed string in holographic background. We then describe the derivation of the Casimir energy for a static case with massive endpoints in . The Liouville or Polchinski-Strominger terms for quantizing the string in non-critical dimension is discussed in . Phenomenology: Comparison between the stringy models and experimental data is reviewed in section . We first describe the fitting models and procedure and then present separately the meson trajectory fits and the baryon trajectory fits and finally we describe the search of glueballs . We summarize and describe several future directions in
2 From to confining string backgrounds
The original duality equivalence is between the SYM theory and string theory in . Obviously the is not the right framework to describe hadrons that should resemble those found in nature. Instead we need stringy dual of four dimensional gauge dynamical system which is non-supersymmetric, non-conformal and confining. The main two requirements on the desired string background is that it (i) admits confinement and (ii) that it includes matter in the fundamental representation invariant under chiral flavor symmetry and that the latter is spontaneously broken.
2.1 Confining background
There are by now several ways to get a string background which is dual to a confining boundary field theory.
An important class of models is achieved by compactifing higher dimensional theories to four dimensions in a way which (partially) breaks supersymmetry. A prototype model of this approach is that of a compactified D5 or NS5 brane on . In an appropriate decoupling limit this provides a dual of the dimensional SYM theory. Another model of this class is the so called Witten’s model  which is based on the compactification of one space coordinate of a D4 brane on a circle. The sub-manifold spanned by the compactified coordinate and the radial coordinate has geometry of a cigar. Imposing on the circle anti-periodic boundary conditions for the fermions, in particular the gauginos, render them massive and hence supersymmetry is broken. In the limit of small compactified radius one finds that the dual field theory is a contaminated low energy effective YM theory in four dimensions. This approach will be described in
It was realized early in the game that one can replace the part of the string background with orbifolds of it or by the conifold. In this way conformal invariance is maintained, however, parts of the supersymmetry can be broken. In particular the dual of the conifold model has instead of . It was shown in  that one can move from the conformal theory to a confining one by deforming the conifold. The corresponding gauge theory is a cascading gauge theory.
In analogy to brane background solutions of the ten dimensional equations of motion  one can find solutions for the metric dilaton and RR forms in non-critical dimensions. In particular there is a six dimensional model of branes compactified on a . This model resembles Witten’s model with the difference that the dual field theory is in the large limit but with ’t Hooft coupling .
The AdS/QCD is a bottom-up approach, namely it not a solution of the ten dimensional equations of motion, based on an gravity background with additional fields residing on it. The idea is to determine the background in such a way that the corresponding dual boundary field theory has properties that resemble those of QCD. The basic model of this kind is the “hard wall” model where the five-dimensional space is truncated at a certain value of the radial direction . The conformal invariance of the space corresponds to the asymptotic free UV region of the gauge theory. Confinement is achieved by the IR hard wall as will be seen . This construction was first introduced in . Since this model does not admit a Regge behaviour for the corresponding meson spectra there was a proposal  to improve the model by “softening” the hard wall into a soft-wall model. It was argued that in this way one finds an excited spectrum of the form but, as will be discussed below also, this model does not describe faithfully the spectra of mesons.
Another AdS/QCD model is the improved holographic QCD model (IHQCD) which is essentially a five-dimensional dilaton-gravity system with a non-trivial dilaton potential. By tuning the latter, it was shown that one can build a model that admits certain properties in accordance with lattice gauge theory results both at zero and finite temperature.
2.2 Introducing fundamental quarks
Since the early days of string theory it has been understood that fundamental quarks should correspond to open strings. In the modern era of closed string backgrounds this obviously calls for D branes. It is thus natural to wonder, whether one can consistently add D brane probes to supergravity backgrounds duals of confining gauge theories, which will play the role of fundamental quarks. In case that the number of D brane probes one can convincingly argue that the back reaction of the probe branes on the bulk geometry is negligible. It was also well known that open strings between parallel D7 branes and D3 branes play the role of flavored quarks in the gauge theory that resides on the D3 4d world volume. Karch and Katz  proposed to elevate this brane configuration into a supergravity background by introducing D7 probe branes into the background. Note that these types of setups are non-confining ones. This idea was further explored in  and in many other followup works. For a list of them see the review . There have been certain attempts to go beyond the flavor probe approximation for instance . The first attempt to incorporate flavor branes in a confining background was in  where D7 and anti-D7 branes were introduced to the Klebanov-Strassler model . This project was later completed in . An easier model of incorporating and anti branes will be discussed in the next subsection. Flavor was also introduced in the bottom-up models confining models. For instance the Veneziano limit of large in addition to the large was studied in .
2.3 Review of the Witten-Sakai-Sugimoto model
Rather than describing a general confining background, or the whole list of such models, we have chosen a prototype model, the Sakai Sugimoto model, which will be described in details in the next subsection.
Witten’s model  describes the near-horizon limit of a configuration of large number of D4-branes, compactified on a circle in the direction ( see figure 1) with anti-periodic boundary conditions for the fermions.
This breaks supersymmetry and renders only the gauge field to remain massless whereas the gauginos and scalars become massive. To incorporate flavor this model is then elevated to the so called Sakai Sugimoto model  to which a stack of D8-branes (located at ) and a stack of anti-D8-branes (located at the asymptotic location ) is added. This is dual to a dimensional maximally supersymmetric Yang-Mills theory (with coupling constant and with a specific UV completion that will not be important for us), compactified on a circle of radius with anti-periodic boundary conditions for the fermions, with left-handed quarks located at and right-handed quarks located at (obviously we can assume ).
In the limit , this background can be described by probe D8-branes inserted into the near-horizon limit of a set of D4-branes compactified on a circle with anti-periodic boundary conditions for the fermions. This background is simply related to the (near-horizon limit of the) background of near-extremal D4-branes by exchanging the roles (and signatures) of the time direction and of one of the spatial directions. Let us now briefly review this model, emphasizing the manifestations of confinement and chiral symmetry breaking. The background of type IIA string theory is characterized by the metric, the RR four-form and a dilaton given by
where is the time direction and () are the uncompactified world-volume coordinates of the D4 branes, is a compactified direction of the D4-brane world-volume which is transverse to the probe D8 branes, the volume of the unit four-sphere is denoted by and the corresponding volume form by , is the string length and finally is a parameter related to the string coupling. The submanifold of the background spanned by and has the topology of a cigar (as in both sides of figure 2 below) where the minimum value of at the tip of the cigar is . The tip of the cigar is non-singular if and only if the periodicity of is
and we identify this with the periodicity of the circle that the -dimensional gauge theory lives on.
The parameters of this gauge theory, the five-dimensional gauge coupling , the low-energy four-dimensional gauge coupling , the glueball mass scale , and the string tension are determined from the background (1) in the following form :
where , is the typical scale of the glueball masses computed from the spectrum of excitations around (1), and is the confining string tension in this model (given by the tension of a fundamental string stretched at where its energy is minimized). The gravity approximation is valid whenever , otherwise the curvature at becomes large. Note that as usual in gravity approximations of confining gauge theories, the string tension is much larger than the glueball mass scale in this limit. At very large values of the dilaton becomes large, but this happens at values which are of order (in the large limit with fixed ), so this will play no role in the large limit that we will be interested in. The Wilson line of this gauge theory (before putting in the D8-branes) admits an area law behavior , which means a confining behavior, as can be easily seen using the conditions for confinement of .
Naively, at energies lower than the Kaluza-Klein scale the dual gauge theory is effectively four dimensional; however, it turns out that the theory confines and develops a mass gap of order , so (in the regime where the gravity approximation is valid) there is no real separation between the confined four-dimensional fields and the higher Kaluza-Klein modes on the circle. As discussed in , in the opposite limit of , the theory approaches the dimensional pure Yang-Mills theory at energies small compared to , since in this limit the scale of the mass gap is exponentially small compared to . It is believed that there is no phase transition when varying between the gravity regime and the pure Yang-Mills regime, but it is not clear how to check this.
Next, we introduce the probe 8-branes which span the coordinates , and follow some curve in the -plane. Near the boundary at we want to have D8-branes localized at and anti-D8-branes (or D8-branes with an opposite orientation) localized at . Naively one might think that the D8-branes and anti-D8-branes would go into the interior of the space and stay disconnected; however, these 8-branes do not have anywhere to end in the background (1), so the form of must be such that the D8-branes smoothly connect to the anti-D8-branes (namely, must go to infinity at and at , and must vanish at some minimal coordinate ). Such a configuration spontaneously breaks the chiral symmetry from the symmetry group which is visible at large , , to the diagonal symmetry. Thus, in this configuration the topology forces a breaking of the chiral symmetry; this is not too surprising since chiral symmetry breaking at large follows from rather simple considerations. The most important feature of this solution is the fact that the D8 branes smoothly connect to the anti D8 branes.
In order to find the 8-brane configuration, we need the induced metric on the D8-branes, which is
where . It is easy to check that the Chern-Simons (CS) term in the D8-brane action does not affect the solution of the equations of motion. More precisely, the equation of motion of the gauge field has a classical solution of a vanishing gauge field, since the CS term includes terms of the form and . So, we are left only with the Dirac-Born-Infeld (DBI) action. Substituting the determinant of the induced metric and the dilaton into the DBI action, we obtain (ignoring the factor of which multiplies all the D8-brane actions that we will write) :
where is the induced metric (6) and includes the outcome of the integration over all the coordinates apart from . From this action it is straightforward to determine the profile of the D8 probe branes. The form of this profile which is drawn in figure 2(a) is given by
where . Small values of correspond to large values of . In this limit we have leading to . For general values of the dependence of on is more complicated.
There is a simple special case of the above solutions, which occurs when , namely the D8-branes and anti-D8-branes lie at antipodal points of the circle. In this case the solution for the branes is simply and , with the two branches meeting smoothly at the minimal value to join the D8-branes and the anti-D8-branes together (see figure 1). This is the solution advocated in . The generalized, not necessarily antipodal configuration that was described above was introduced in . As will be discussed in section the difference between the antipodal and the non-antipodal will translate to the difference between stringy meson with massless versus massive endpoints. It was also found out in  that, in order to have attraction between flavor instantons that mimic the baryons, the setup has to be non-antipodal. In the approach of considering the duals of the mesons as fluctuation modes of the flavor branes, and not as we argue here in this review as string configuration, it turned out that the difference between and is not the dual of the QCD quark masses. One manifestation of this is the fact that the Goldstone bosons associated with the breakdown of the chiral flavor symmetry remain massless even for the non-antipodal case. This led to certain generalizations of the Sakai Sugimoto model by introducing additional adjoint “tachyonic”’ field into the bulk ,,  and by introducing an open Wilson line. In view of the HISH these type of generalization are obviously not necessary.
This type of antipodal solution is drawn in figure 2(b).
It was shown in  that the classical configurations both the antipodal and the non-antipodal are stable. This was done by a perturbative analysis (in ) of the backreaction of the localized D8 branes. The explicit expressions of the backreacted metric, dilaton and RR form were written down and it was found that the backreaction remains small up to a radial value of ), and that the background functions are smooth except at the D8 sources. In this perturbative window, the original embedding remains a solution to the equations of motion. Furthermore, the fluctuations around the original embedding, do not become tachyonic due to the backreaction in the perturbative regime. This is is due to a cancellation between the DBI and CS parts of the D8 brane action in the perturbed background. For further discussion of the pros and cons of the Sakai-Sugimoto model see .
The main results reviewed in this subsection hold also to an analogous non-critical setup  based on inserting D4 flavor branes into the background of compactified colored D4 branes and in particular the structure of the spontaneous breaking of the flavor chiral symmetry.
3 Hadrons as strings in holographic background
In this section we will analyze various classical string configurations in confining holographic models. In fact we will define below what is a confining string background according to the behavior of the classical static string that attaches its boundary. We will treat four types of strings:
(i) The string dual of a Wilson line.
(ii) A rotating open string that is attached to flavor branes. This will be the dual of a meson.
(iii) A rotating configuration of strings attached to a “baryonic vertex”, the dual of the baryon.
(iv) A rotating folded closed string, the dual of the glueball.
3.1 The holographic Wilson line
One of the most important characteristics of non-abelian gauge theories, in particular the ones associated with the group , is the Wilson line which is a non-local gauge invariant expectation value that takes the form
where denotes path ordering and is some given contour.
For the special case where is a strip of length along one space direction and along the time direction, one can extract from the corresponding Wilson line, in Euclidean space-time and for , the potential between a quark and an anti-quark as follows
The holographic dual of the expectation value of the Wilson line which was determined in  is given by
where is the renormalized Nambu-Goto action of a string worldsheet whose boundary on the boundary of the bulk space-time is . There are several methods of renormalizing the result. In particular in the one we discuss below one subtracts the infinite action of the straight strings which are duals of the masses of quark anti-quark pair.
Next we would like to apply the holographic prescription of computing the Wilson line to a particular class of holographic models. The latter are characterized by higher than five dimensional space-time with a boundary. The coordinates of these space-times include the coordinates of the boundary space-time, a radial coordinate and additional coordinates transverse to the boundary and to the radial direction. We assume that the corresponding metric depends only on the radial coordinate such that its general form is
where is the time direction, is the radial coordinate, are the space coordinates on the boundary and are the transverse coordinates. We adopt the notation in which the radial coordinate is positive defined and the boundary is located at . In addition, a “horizon” may exist at , such that spacetime is defined in the region , instead of as in the case where no horizon is present.
The construction we examine is that of the strip discussed above. This takes the form of an open string living in the bulk of the space with both its ends tied to the boundary. From the viewpoint of the field theory living on the boundary the endpoints of the string are the pair, so the energy of the string is related to the energy of the pair. In order to calculate the energy of this configuration on the classical level we use the notations and results of . First, we define
Upon choosing the worldsheet coordinates ( is a coordinate on the boundary pointing in the direction from one endpoint of the string to the other one) and . This is obviously the most natural gauge choice for the case of the strip Wilson line. If in addition we assume translation invariance along , the Nambu-Goto action describing the string takes the form
where we assumes a static configuration, is the length of the strip along the time direction and is the string tension which from here on we set to be one. Later we will bring it back again. Letting the derivative with respect to , , playing the role of the time derivative in standard canonical procedure, then the conjugate momentum and the Hamiltonian are
As the Hamiltonian does not depend explicitly on , its value is a constant of motion. We shall deal with the case in which is an even function, and therefore there is a minimal value for which At that point we see from (17) that . The constant of motion is therefore
from which we can extract the differential equation of the geodesic line
and re-express the on-shell Lagrangian (i.e. the Lagrangian on the equation of motion) as a function of only
Then the distance between the string’s endpoints (or the distance between the “quarks”) is
where is the minimal value in the radial direction to which the string reaches and is the value of on the boundary. The bare energy of the string is given by
Generically, the bare energy diverges and hence a renormalization
procedure is needed. There are several renormalization schemes to deal with this infinity. These were summarized in . Here we follow 
and use the mass subtraction scheme in which the bare masses of the
quarks are subtracted from the bare energy. The bare external quark mass is
viewed as a straight string with a constant value of , stretching
from (or if there exists an horizon at )
to , such that it is given by
Then the renormalized energy would be given by
It is important to emphasize that this result for the energy is only at the classical level and does not include quantum corrections.
In order to reproduce the QCD heavy quarks potential we first have to demand the holographic models to reproduce the asymptotic forms of the potential. This leads to several restrictions on the forms of the and functions. The condition for confining behavior at large distances was derived in :
has a minimum at and or
diverges at and
Then the string tension is given by or , correspondingly. The second asymptotic is perturbative QCD at small distances. The conditions on the background to reproduce the leading perturbative behavior of QCD, which is Coulomb-like, were derived in  .
The physical picture arising from this construction is as follows. The confining limit is approached as , then most of the string lies at the vicinity of which implies in the dual field theory a string-like interaction between the “quarks” with a string tension (upon bringing back that was set above to one)
and a linear potential
where is a finite constant. On the other hand, the Coulomb-like limit is approached as , then the whole string is far away from and is ruled by the geometry near the boundary. The conditions for such a behavior were analyzed in .
’t Hooft line
In addition to the Wilson line one discusses also the Polyakov line and the ’t Hooft line. The former is a Wilson line for which the contour is the Euclidean time direction compactified on a circle. Correspondingly the holographic string realization of the Polyakov loop is a string whose worldsheet wraps the compactified time direction. We will not discuss it here. In YM theories the “electric-magnetic” dual of the Wilson line is the ’t Hooft line. Just as one extracts the quark anti-quark potential from the former with the strip contour, the later determines for the same contour the monopole anti-monopole potential.
We would like to address now the question of how to construct a string configuration which associates with the ’t Hooft line. The answer for that question is that it should be a configuration of a brane starting and ending on the boundary. The prototype confining background described in is that of a compactified D4 branes. This is a type II-A theory where the allowed D branes are only of the form and hence (at least in the supersymmetric case) one cannot embed a D1 brane. Instead we will take the stringy realization of the ’t Hooft line as a D2-brane ending on the D4-brane. The D2-brane is wrapped along the compact space direction so from the point of view of the four dimensional theory it is a point like object. The DBI action of a D2-brane is
where is the world-volume induced metric and is the dilaton. For the D2-brane we consider, which is infinite along one direction ( we denote its length by and winds the direction), we get
Note the factor which is expected for a monopole. The distance between the monopole and the anti-monopole is
where . The energy (after subtracting the energy corresponding to a free monopole and anti-monopole) is
For it is energetically favorable for the system to be in a configuration of two parallel D2-branes ending on the horizon and wrapping . So in the ”YM region” we find screening of the magnetic charge which is another indication to confinement of the electric charge. For further details on the holographic determination of the ’t Hooft line see for instance .
3.2 The stringy duals of mesons
Next we want to identify among the strings that can reside in a holographic backgrounds those that correspond to mesons. Naturally a meson associates with an open string and the quark and anti-quark that it is built from with the string endpoints. As we have seen in the previous section, endpoints on the boundary of the holographic bulk correspond to infinitely heavy external quarks. Holographic backgrounds are associated with type II closed string theories. In such backgrounds open strings can start and end only on D-branes (or the boundary). Thus “dynamical quarks” should be related to string endpoints which are not on the boundary but rather they attach to D-branes. As was discussed in section flavored confining backgrounds include stacks of D branes which are referred to as “flavor branes”. We consider only the case of , where is the number of branes that constitute in the near horizon limit the holographic background. Since the endpoints can freely move, then classically the strings do not shrink to zero size only if they rotate and the “centrifugal force” is balancing the string tension. Thus, to describe the stringy meson we will now discuss (i) The conditions on strings rotating at constant radial coordinate. (ii) Rotating strings attached to flavor branes.
The conditions on strings rotating at constant radial coordinate
Consider the background metric of (14). Setting aside the transverse coordinates, the metric reads
where here we denote the metric along the space direction as , instead of as it was denoted in (14). The metric as well as any other fields of the background depend only on the radial direction . Since we have in mind addressing spinning strings, it is convenient to describe the space part of the metric as
where is the direction perpendicular to the plane of rotation. The classical equations of motion of a bosonic string defined on this background can be formulated on equal footing in the NG formulation or the Polyakov action. Let us use now the latter. The equations of motion associated with the variation of and respectively are
where denotes the worldsheet coordinates and . In addition in the Polyakov formulation one has to add the Virasoro constraint
where , and stands for the contribution to the Virasoro constraint of the rest of the background metric.
Next we would like to find solutions of the equations of motion which describe strings spinning in space-time. For that purpose we take the following ansatz
It is obvious that this ansatz solves the first two equations. The third equation together with the Virasoro constraint is solved (for the case that ) by with . The boundary conditions we want to impose will select the particular combination of and . Let us now investigate the equation of motion associated with and for the particular ansatz . This can be a solution only provided
This is just the first condition for having a confining background. The condition insures that the Virasoro constraint is obeyed in a nontrivial manner.
We would like to check the condition in the class of confining models that was discussed in namely those of compactified D branes and in particular the model of the near extremal brane, compactified on an . Denoting the direction along the denoted by , the corresponding component of the metric is related to that of the direction as . Thus the condition for having a solution of the equation of motion with includes the condition
This condition is obeyed if at , which is the second condition for a confining background with , and with the the demand of non-vanishing to have a non-trivial Virasoro constraint.
To summarize, we have just realized that there is a close relation between the conditions of having area law Wilson loop and of having a spinning string configuration at a constant radial coordinate.
The meson as a rotating string attached to flavor branes
We have seen above the conditions for having a rotating string at constant . We would like to examine whether rotating strings with endpoints attached to flavor branes can obey these conditions. For the Wilson line in a “confining background” we found a string configuration that stretches vertically from the boundary down to the vicinity of the “wall”’, then flattens along the wall and then “climbs” up vertically to a flavor brane. In a very similar manner we search for a string of the same type of profile but now not a static one. When the endpoints of the open string are not on the boundary and are not nailed down, the string does not shrink due to the tension only if it rotates and the “centrifugal force” balances the tension. Therefore we look for a solution of the string equation of motion of the form of a rotating string in a plane spanned by the coordinates. Like the stringy Wilson line, the rotating string in the holographic background will include two vertical segments (region I) which connect the horizontal segment of the string (region II) to the flavor branes. Thus, the procedure to determine the holographic dual of the meson includes a solution of the equation of motion in the vertical segment (region I), a solution in the horizontal segment (region II), matching the two solutions and finally computing the corresponding energy and angular momentum as the Noether charges associated with the translation symmetry along the time and azimuthal directions. Before delving into the equations it should be useful to have a picture of the “ mesonic rotating string”. It appears in figure (5).
The Nambu-Goto action associated with a rotating string in a background metric of the form (33) takes the form
where we assume here that and where and are defined in (15). Since we have in mind analysing rotating strings we do not consider the most general configuration but rather an ansatz similar to the one discussed above (41)
The boundary conditions corresponding to this configuration are Neumann in the directions parallel to the flavor brane, and Dirichlet in the directions transverse to the brane, namely:
In fact for the rotating string we have and since we can use the static gauge where the action in fact reads
The equation of motion with respect to is given by
Let us now examine whether this equation (49) indeed admits a solution of an horizontal part (region II) connected to flavor branes by vertical parts (region I). In regions I that are along
where along these segments with is the length of the string. Thus the l.h.s of the equation turns into which is also the value of the r.h.s. In region II which is along
since we require there that is ( approximately) constant we expand the equation (49) to leading order in . For finite the l.h.s of the equation vanishes and thus since in this case it implies that has to have an extremum at . The other option that we want to examine is the case that which for the static case is one of the two sufficient conditions  to admit an confining area law behavior.
Since we took the string to stretch separately along region of the type I, region II and again region of the type I we have to reexamine the variation of the action with respect to .
Upon substituting the solution of the bulk equations of motion and in particular that in the interval , we find that the variation of the action can vanish only provided that
Let us define now the notion of a string endpoint mass
then the condition of the vanishing of the variation of the action takes the form
As will be further discussed in section this is nothing but a balancing equation between the tension and the centrifugal force acting on the string endpoint, where the tension is exactly the same one that was found above for the Wilson line.
Next we determine the energy and angular momentum of the rotating string that get contributions from both the horizontal and vertical segments. The energy and angular momentum are the Noether charges associated with the invariance of the action under shifts of and respectively, given by
The contribution of regions I is therefore given by
which as will be discussed in section maps into the contribution to the energy and angular momentum of the massive endpoints of the string. The contribution of the string that stretches along region II is
Combining together the contributions of regions I and region II we find
So far we have considered stringy holographic baryons that attach to one flavor brane. In holographic backgrounds one can introduce flavor branes at different radial locations thus corresponding to different string endpoint masses, or different quark masses. For instance, a setup that corresponds to and quarks of the same mass, a strange quark, a charm quark, and a bottom quark is schematically drawn in figure (6). A meson composed of a bottom quark and a light anti-quark was added to the figure.
3.3 Holographic stringy baryons
We have seen in the previous sections that (i) a string with its two ends on the boundary of a holographic background corresponds in the dual field theory to an external quark, and (ii) a rotating string with its ends on flavor probe branes is a dual of a meson. It is thus clear that a stringy holographic baryon has to include strings that are connected together and end on flavor branes (dynamical baryon) or on the boundary (external baryon). The question is what provides the “baryonic vertex” that connects together strings. In  it was shown that in the background, which is equipped with an RR flux of value , a brane that wraps the has to have strings attach to it. This property can be generalized to other holographic backgrounds so that a