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Chiral Dynamics From AdS Space

We study the low energy dynamics of pions in a gravity dual of chiral symmetry breaking. The string theory construction consists of a probe D7 brane in the Constable Myers non-supersymmetric background, which has been shown to describe chiral symmetry breaking in the pattern of QCD. We expand the D7 brane's Dirac Born Infeld action for fluctuations that correspond to the Goldstone mode and show that they take the form of a non-linear chiral lagrangian. We numerically compute the quark condensate, pion decay constant and higher order Gasser Leutwyler coefficients. We find their form is consistent with naive dimensional analysis estimates. We also explore the gauging of the quark's chiral symmetries and the vector meson spectrum.


Introduction
The dynamics and phenomenology of QCD are dominated by quarks. In particular the vast majority of known hadronic states can be identified as having constituent quarks and the low energy dynamics is controlled by the chiral symmetry breaking quark condensate. The discovery of the AdS/CFT Correspondence [1,2,3] has raised the hope of providing a weakly coupled gravitational description of the strong coupling phase of QCD. Understanding fundamental representation quarks in this setting must therefore be a priority.
Recently a simple mechanism for including quarks in the AdS/CFT Correspondence has been found by Karch and Katz [19]. A probe D7 brane is added to the original D3 brane construction of the AdS/CFT Correspondence. The new "37" open strings generate quarks in the field theory on the D3 brane world volume. The world volume theory has N = 2 supersymmetry. Karch and Katz identified the additional "77" open strings on the D7 world volume as playing a holographically dual role to the gauge invariant quark operators of the gauge theory. Treating the D7 as a probe corresponds to quenching in the gauge theory. The dynamics of quarks and their bound states have been studied in a number of supersymmetric gauge field backgrounds using these techniques [20] - [28].
If we wish to study chiral symmetry breaking in the pattern of QCD we must look at a non-supersymmetric gauge theory since supersymmetry forbids such a quark condensate. A number of gravitational duals of non-supersymmetric backgrounds exist [5] - [9] and [12]- [18]. The simplest cases involve the deformation of the original AdS/CFT Correspondence by the inclusion of relevant operators in the field theory [4], which corresponds to switching on bulk supergravity fields. In these cases the UV of the gauge theory returns to N = 4 super Yang Mills and the operator identification between the two dual theories is cleanly understood. Such theories do not have total decoupling of the super-partners that are given mass since the theory is strongly coupled in the UV -these extra states have masses of order the strong interaction scale Λ. Simply breaking supersymmetry is sufficient to allow a quark condensate though. Nonsupersymmetric deformations [12]- [18] of the Klebanov Strassler [10] and Maldecena Nunez [11] gravity duals also exist but here the operator matching, even in the UV, is less clear.
The first study of chiral symmetry breaking in this formalism was made in [22]. The deformed AdS 5 × S 5 geometry of Constable and Myers [9], which corresponds to the addition of an Rchargeless dimension 4 operator such as TrF µν F µν to the N = 4 theory, was used. The existence of a condensate and massless pions in the limit where the quark mass went to zero were identified. In this paper we will make further study of that case. The Constable Myers' geometry has a singularity in the interior, the precise significance of which is unclear; the singularity might correspond to the presence of some expanded D-brane set up in the interior and signal the strong interaction scale Λ. It turns out that the core of the geometry is repulsive to the D7 brane probe when quarks are included and this is what triggers the chiral symmetry breaking in the model. Pleasingly this repulsion also makes sure that the D7 branes avoid the central singularity so for the purposes of this study we can set aside study of the singularity. Chiral symmetry breaking by the same mechanism has also been studied in [23]. They use a geometry around D4 branes wrapped on a circle which describes a 3+1 dimensional Yang Mills like theory in the IR. This geometry has no interior singularity but the core is again repulsive to D7 brane probes triggering chiral symmetry breaking. In this case though the UV of the theory becomes strongly coupled and wishes to become an M5 brane construction. The universality of the IR mechanism is encouraging and provides support for further study in the Constable Myers background.
Here we will return to that Constable Myers configuration [9,22] and examine chiral symmetry breaking and its consequences in more detail. First we refine the numerical analysis of [22] by solving the Euler Lagrange equation, describing how the D7 brane lies in the geometry, starting with the appropriate regular infra-red boundary conditions. This allows us to identify the regular physical flows without tuning. We stress the geometrical description of chiral symmetry breaking provided by the set up where the repulsion of the interior geometry forces the D7 brane to lie in a symmetry breaking configuration. For small quark mass the vacuum energy of the configuration is proportional to the quark mass as expected in the chiral lagrangian formalism [29,30] (which we will review in the next section). It is therefore possible to extract the quark condensate which we show matches with the value obtained in [22] by looking at the UV boundary conditions on the flows.
Next we move on to study fluctuations of the D7 about the vacuum configuration, that describe the Goldstone mode, or pion fields. We show that the lagrangian terms match those expected in the chiral lagrangian and then compute the couplings. In particular the pion mass is shown numerically to have a linear dependence on the square root of the quark mass for low quark mass. It is then possible to compute the pion decay constant which we show has the correct dependence on the number of colours N and has a numerical suppression factor relative to the strong coupling scale consistent with that seen in QCD and naive arguments [29,30,31].
In the chiral limit we then look at terms in the low energy lagrangian involving four pion fields and estimate their size. We again find a match to naive estimates [32,31].
A natural next step is to include multiple D7 probes and study the resulting theory with N f quark flavours via the non-abelian DBI action [34]. In fact the quarks couple to the adjoint scalar field in the N = 2 UV gauge theory via a superpotential termQAQ and enhancing the number of quark flavours does not therefore enhance the chiral flavour group. The adjoint scalar is though massive on the scale Λ and we might expect an accidental symmetry in the IR. This phenomena has been discussed in the context of [23]. The additional fields in the non-abelian DBI action are also massless but have interaction terms that are not Goldstone like -these are just the remnants of the usual commutator interactions of the scalar fields on a brane. As a test of the Goldstone like nature of these fields we compute the pion decay constant via lagrangian terms that are not present in the abelian case, neglecting the commutator interactions. We find excellent numerical agreement with our previous value showing that these fields are rather Goldstone like.
Finally we study the gauge field on the world volume of the D7 which is dual to weakly gauging the vector U(1) baryon number symmetry and also the vector meson spectrum. We have been unable to identify the fields associated with the axial vector mesons though, which, for example, stops us testing vector meson dominance in the Weinberg sum rules [36]. Nevertheless we compare the vector meson spectrum to that of the N = 2 theory provided by a pure AdS background.

Chiral Dynamics
QCD with N f massless quarks has a chiral SU(N f ) L × SU(N f ) R global symmetry. When asymptotic freedom drives the coupling strong it is believed that this symmetry is broken by a quark bilinear condensate to the vector SU(N f ) subgroup. The symmetry breaking produces N 2 f − 1 Goldstone bosons (N 2 f at large N c where instanton effects are suppressed) which are associated with the pion multiplet in nature. Since the quarks have small current masses the pions are only pseudo-Goldstone fields. The very low energy dynamics only involves these Goldstone fields and can be completely described by a theory that realizes the broken symmetry non-linearly [29,30]. Such a phenomenological theory is called a chiral lagrangian. We introduce fields Π a (with T a the generators of the broken SU(N f ) group normalized such that TrT a T b = 1 2 δ ab ) where f Π is the pion decay constant and where the chiral symmetry transformation properties are shown. Remembering that the quark mass matrix transforms as M → L † MR we can then write a lagrangian at leading order in both a derivative expansion and an expansion in expanding U to find the leading terms for the pion fields gives The couplings f Π etc must be found phenomenologically in the low energy theory but are in principle predictions of the full high energy QCD theory. Since M is a source for the quark bilinear condensate the coupling ν is related to the value of the quark condensate Note the pion mass can then be written as the Gell-Mann-Oakes-Renner relation m 2 Π = 2mq qq The interaction terms in the low energy theory at next order in the chiral expansion have been written down by Gasser and Leutwyler [32] and can be parameterized as We will be interested in showing that the pions of our gravity construction conform to this structure. We will also estimate some of these coefficients below using the AdS/CFT Correspondence method.

Naive Dimensional Analysis
A simple set of rules for estimating the size of chiral Lagrangian couplings has been devised [31].
In QCD there are two scales -Λ the strong coupling scale generated by the running coupling, and the pion decay constant, f π , which is approximately Naive dimensional analysis says that one should give all chiral lagrangian terms a common coefficient of Λ 2 f 2 Π with any occurrences of M or D µ being dimensionally balanced by a factor of Λ. Note that the pion fields enter in U dimensionally balanced by f Π . For example the L i coefficients are predicted to be of order 1/16π 2 using these rules which reasonably matches their physical values.
One of our goals in this paper is to test this naive power counting in the strongly coupled gauge theory for which we have a gravitational dual. We will find reasonable agreement below.

The Brane Construction and Gravity Dual
AdS/CFT Correspondence type duals are obtained by equating the physics on a stack of N coincident, flat D3 branes and the supergravity background dynamically generated by the D3 branes' tension. We will consider the non-supersymmetric deformed AdS geometry originally constructed in [9]. This geometry is dual to the N = 4 super Yang-Mills theory (with g 2 Y M = 2πg s ) deformed by the presence of a vacuum expectation value for an R-singlet operator with dimension four (such as T rF µν F µν ). The supergravity background has a dilaton and S 5 volume factor depending on the radial direction. The geometry has a naked singularity in the interior which we loosely expect to correspond to the presence of the central stack of D3 branes. Whether this geometry actually describes the stable non-supersymmetric vacuum of a field theory is not well understood [9]. This is not so important from our point of view though since the geometry is a well defined gravity description of a non-supersymmetric gauge configuration. We will just study the behaviour of quarks in that background.
The geometry in Einstein frame is given by where and the dilaton and four-form become There are formally two free parameters, R and b, since As usual in the AdS/CFT Correspondence the w directions have the conformal scaling properties of energy scale in the field theory. Thus b is the only object that breaks the conformal (and super) symmetry of the gauge theory. We define an associated mass scale which is the mass of a string of length b. Since Λ b is the scale of conformal symmetry breaking and the theory is strongly coupled at that scale we expect the dynamical strong coupling scale of the theory, Λ, to lie close to Λ b . We will hence forth loosely associate the two. In fact we are not interested in changing the scale Λ b since it is the only mass scale so we can set b = 1 below. First though lets consider the R dependence of the solution.
The parameter R determines g 2 Y M N in the field theory as usual in the Correspondence . We find it easiest to track the R dependence by writing w and b in units of R so that δ = 1/2b 4 . This means that our fundamental energy scale Λ b scales with R so we should express masses as a ratio of Now we can set b = 1 and the metric we will use is where We will now introduce one flavour of quark via a D7 brane probe in the geometry. The D7 brane lies in the x 4 directions and w 1 − w 4 (it is convenient to define a coordinate ρ such that . This configuration in pure AdS preserves N = 2 supersymmetry on the D3 world volume so corresponds to introducing a quark hypermultiplet Q andQ. N = 2 supersymmetry will ensure there is a superpotential coupling to one of the three adjoint chiral multiplets A of the original N = 4 theory (W =QAQ). In the non-supersymmetric field theory we expect the scalar fields to gain masses of order the supersymmetry breaking scale. The remaining fermionic terms in the gauge theory are of the form where one should note that g s enters the normalization as for the gauge fields (this is not the standard normalization for comparison with eg eq (4) above).
The Dirac Born Infeld action for the probe is where P indicates the pullback of the space-time metric onto the D7 world volume. This action will determine how the D7 lies in the remaining w 5 − w 6 directions. We will use a complex coordinate in this plane Asymptotically where the geometry returns to AdS the resulting Euler Lagrange equation is and has solutions The two integration constants correspond to a mass and vev for the quark bilinearqq with Note that we are writing Φ in units of R and hence it has the same N scaling as the parameter b. These measures of the quark mass and condensate when expressed as multiples of Λ b do not scale with N.
In the massless limit where the D7 brane lives at Φ = 0 there is a U(1) symmetry acting in the Φ plane. Clearly this corresponds to an angular rotation onqq and is hence the U(1) A symmetry of the quarks. In fact this symmetry is also part of the isometries of the space transverse to the central D3 brane stack and is thus part of the SO(6) R symmetry of the gauge background. This reflects the presence of the UV superpotential term W =QAQ which mixes the axial and R symmetries. The A field is hopefully somewhat massive in this configuration (of order Λ b ) but the symmetry is nevertheless this mixture.

Chiral Symmetry Breaking
Let us begin by considering the massless quark limit where there is a good U(1) A symmetry. Asymptotically in the UV the D7 brane lies at Φ = 0. The equation of motion that determines how it lies in the interior is given by where The final terms in the equation of motion is a "potential" like term that is evaluated to be The equation of motion has an explicit Φ → e iα Φ symmetry. In the massless case this is the U(1) A symmetry on the quarks. The solutions to the equation of motion in the interior simplify to the D7 brane sitting at a fixed angle in the plane, θ, with the radial behaviour given by setting Φ = σ real in the equation of motion.
There are two regular solutions of the equation of motion with UV boundary condition σ = c/ρ 2 , shown in Figure 1. They have c = ±1.86 showing that the solution indeed prefers the formation of a chiral symmetry breaking condensate. Note that in the IR (small ρ) these solutions are σ = constant, so they can also be found by numerically solving up from the IR and matching to the UV boundary conditions. This is the simplest way to find solutions since any IR boundary condition of this sort produces a regular flow and it is then not necessary to tune onto the regular flows. The two solutions are just two opposing points on a circle in Φ verifying that there is indeed a set of solutions with the same radial behaviour at each value of Φ. This circle of degenerate solutions is the vacuum manifold. It is interesting to see how the chiral symmetry breaking manifests itself geometrically. The solution for the position of the D7 brane is sketched in Figure 1. Asymptotically it lies at the origin of the plane but in the interior the D7 brane is repelled by the core of the geometry forcing it out into the plane where it breaks the U(1) symmetry.

Massive Case
It is straightforward to also introduce a mass into the UV boundary conditions for Φ. Let us first look at solutions of the form θ = 0 and σ = m + c/ρ 2 in the UV. These will turn out to describe the true vacuum of the theory. The regular solutions have σ tend to a constant in the IR so are easily found by flowing up from the IR. The mass and condensate can then be read from the solution in the UV. We plot a few of these solutions in Figure 2. This procedure also allows us to determine the mass dependence of the condensate and we plot that also in figure 2. It is also interesting to study the vacua with a real mass term but with a phase on the condensate -these will correspond to the vacua around the circle in θ in the m = 0 limit and we expect their energy to be lifted relative to the true vacuum when the mass is non-zero. To study these we use boundary conditions on the UV fields In general we are looking for a regular solution for both w 5 , w 6 and this is very hard to achieve numerically. As an example of evidence for the existence of these solutions though we show in Figure   We can now begin to determine the chiral Lagrangian parameters predicted by this model. We can calculate the vacuum energy as a function of the quark mass for the positive and negative condensate solutions. The vacuum energy is given by the DBI action The angular integral over the S 3 the D7 wraps gives 2π 2 . The resulting four dimensional cosmological term, Ω 4 , should be normalized by where the integral I 0 is over the solutions described above. Note that the factor of g s shows us that the vacuum energy scales as N in the large N limit with g s N kept constant. This is consistent with expectations for the vacuum energy contributions from fundamental quarks.
Asymptotically Ω 4 ≃ ρ 4 U V indicating the expected UV divergence of the vacuum energy. This will be present for all our configurations no matter the quark mass or condensate. We are interested in the corrections due to the low energy chiral dynamics (those in eq(3)) so subtract the massless result for Ω 4 from the vacuum energy for each configuration. We show numerical results for Ω 4 /Λ 4 b in Figure 5. The positive condensate solutions are energetically favourable. This is because the potential for the quark condensate has a tilted wine bottle shape of the form shown in Figure 6. There we sketch a potential of the form V = α|Φ| 4 − β 2 |Φ| 2 + mRe(Φ) -as m increases the potential tilts giving a single true vacuum. In this simple model there is a critical value of m where the central "hump" disappears and there is a single unique vacuum. We have numerically looked for such a solution in the D7 brane case. As the boundary condition on m → 1.5 the solution for the meta-stable vacuum (of the type shown in Figure 4) has a singular derivative as it passes through the w 6 axis and we loose numerical control. This may well be an indication of the absence of such solutions for larger m corresponding to the loss of the hump in the naive model.
Finally we can use the vacuum energy to compute ν 3 in eq (3) and to provide an alternative identification of the quark condensate in the model. The chiral lagrangian predicts that the vacuum energy will be given by Before extracting qq in this way we must normalize the quark fields in the standard way appropriate for these equations. This removes the factor of 1/2πg s in eq (27) and we then extract the condensate from the slope of the vacuum energy vs mass plot. We find numerically qq = 1.86 (29)

Pions and Their Interactions
We will now turn to studying the Goldstone bosons of the chiral symmetry breaking we have observed above. So far we have considered the dynamical breaking of the U(1) A symmetry of a single flavour of quark. The Goldstone boson will therefore be the analogue of the η ′ in QCD, which becomes degenerate with the QCD pions in the limits where the quark masses vanish and at large N where the anomaly is suppressed. We will loosely refer to our state as a pion. We will shortly consider extending this construction to non-abelian axial symmetries.

The Pion Mass and f π
The pion in the one flavour case will be a bound state of a quark and an anti-quark and hence will be described by the D7 world volume field Φ. The Goldstone fluctuation corresponds to oscillations of the field along the vacuum manifold -in the brane language this will correspond to fluctuations of the brane in the direction of the possible set of solutions for its background position. In the chiral limit this corresponds to fluctuations in the angular θ direction in the w 5 − w 6 plane. To leading order, if we have a background configuration for the D7 brane described by a particular real solution of (22), σ 0 , we can look at small fluctuations in the θ angular direction We look for solutions where k 2 = M 2 . The action for these fluctuations to quadratic order is given by expanding the DBI action (note that the metric is independent of the angle θ) Note that the DBI action only produces terms with derivatives of θ -we can though obtain potential terms in the pion field when those derivatives are ρ derivatives that act on f (ρ). We find the resulting equation of motion for f (32) This equation can be numerically solved for MR as a function of m using the UV boundary condition f = 1/ρ 2 (reflecting the fact that the pion has the UV scaling dimension ofqq) and seeking regular solutions for f . We plot the result (in which m determines the σ 0 flow) in Figure  7. We indeed find a massless pion at m = 0 in accordance with Goldstone's theorem. Note that in the case where the pion mass vanishes the equation of motion is just the first term above and when substituted back into (31), after integration by parts, explicitly makes the (∂ ρ θ) 2 term vanish. In the chiral limit if one worked to higher order all higher order terms involving (∂ ρ θ) n would also vanish directly. This demonstrates that the vacuum manifold is a truly flat direction.
Below we will only look at terms at higher order involving the spatial derivatives ∂ µ .
In figure 7 we also plot MR vs √ m for small m -there is a good linear fit matching the expectations for the pion mass dependence on m in the chiral lagrangian (3). We will write where we have numerically determined κ = 2.75.
Note that the appropriate four dimensional, low energy action can be found by writing θ = 2πα ′ f (ρ)Π(x) and substituting the equation of motion for f (ρ) back into the action (31), after integrating by parts. This gives where The coefficient at the front of (34) should be absorbed into the normalization of Π to make the kinetic term canonical. We must be careful to include this normalization below. Note also that I 1 is only defined upto the normalization of f which is free in the linearized equation of motion -this factor will cancel when Π is canonically normalized in all quantities below. The chiral Lagrangian predicts that Since we have extracted ν 3 from the vacuum energy above we can now determine f π . With the supersymmetric normalization of fields we found numerically Writing our result for the pion mass in units of Λ b gives and hence we find for f π which has the expected dependence on N and is suppressed relative to Λ b by roughly a factor of 4π 2 . This seems to match well with the naive estimate of section 2.

Higher Order Interactions
It is also interesting to study higher order interaction terms such as those in (5). To do this we can expand the DBI action beyond linearized order in θ and use the linearized solution for f (ρ).
In the chiral limit the identification of the pion field with fluctuations of the brane in the θ direction is rigorous. However, when there is a quark mass present the vacuum will be distorted from a true circle centered on the origin in Φ and this procedure does not correctly identify the pseudo-Goldstone boson (the angular direction will have some of the massive "higgs" like mode mixed in). In principle to find the pseudo Goldstone boson in the massive case one would need to know how to separate the function σ 0 (r) into pieces that represent the running of the mass and the condensate separately. The pseudo Goldstone would correspond to angular fluctuations about just the condensate piece. In other words simply changing the phase on σ 0 changes the phase on the mass which one does not want to do. We have not been able to resolve this so will simply study the massless chiral limit here.
As mentioned above in the chiral limit any terms in the action involving (∂ ρ θ) n vanish by the equations of motion so we will neglect those here (in fact we have checked they explicitly vanish when evaluated on our solution for f (ρ) in this limit). Now we can study the Gasser-Leutwyler coefficients L 1 , L 2 , L 3 . In the chiral Lagrangian, these appear in the form: Since we only have an abelian U(1) symmetry breaking pattern, we can not differentiate between these terms. When U is expanded, each term has a factor of (∂ µ Π) 4 so we can't extract any information about L 1 , L 2 or L 3 separately. It is still interesting to extract the coefficient of the (∂ µ Π) 4 term. This term in the action is Rewriting this in terms of the field Π gives where We must remember to normalize the pion field so the kinetic term in (34) is canonically normalized from which we find the coefficient of the (∂ µ Π) 4 term (which we loosely write as 4L/f 4 π ) is given by (44) or using the expression above for f π This has the same large N scaling (scaling as f 2 Π ) as the naive dimensional analysis estimate. Numerically we find I 2 /I 2 1 = 0.29. If we take g 2 Y M N = (4π) 2 which seems a reasonable strong coupling value then we find L ≃ 0.016N. This is about a factor of 2-3 bigger than N/16π 2 = 0.006N and hence broadly consistent with naive dimensional analysis.

Non-Abelian case
Naively we can introduce extra flavours of quarks by including additional probe D7 branes and using the non-abelian DBI action [34]. As mentioned in the introduction we do not generate a U(N) axial symmetry because of the superpotential termQAQ in the UV theory. The adjoint scalar though is expected to have a mass of order Λ and we might expect the theory to have an accidental chiral symmetry. The non-abelian DBI action requires us to treat the scalar field Φ as a matrix which for the pions will require us to write Where T a are the broken generators of the Lie group. With N f D7 branes this is U(N f ). The DBI action acquires a trace over the flavour indices. As pointed out in [23] if this were the only change then we would find N 2 f copies of the action we had above and hence N 2 f massless pions. The DBI action also contains terms though that generate the usual commutator scalar interactions (tr[φ i , φ j ] 2 ) -these do not though generate a mass for the pions. The explicit chiral symmetry breaking appears therefore to be quite weak in its effects. To pursue this issue further we can study whether the interactions of the pions, neglecting the commutator terms, coincide with expectations from chiral symmetry.
For simplicity we will work in the case with just two flavours and look at just the interactions of two pions, Π 1 and Π 2 . This means that we should write: Note that in the chiral limit we can drop all terms in the DBI action involving ρ derivatives since they will cancel by the equation of motion as discussed above. The resulting action then takes the form of a chiral lagrangian with spatial derivative acting on the field U. We find the lagrangian for the two pion fields takes the form and Performing the equivalent expansion in the chiral lagrangian to compare these terms we find: Note that the relative sizes of the two new interaction terms match between the DBI and chiral lagrangians (in both cases because they result from the expansion of the same exponential form).The two new interaction terms provide an additional opportunity to calculate f Π . Again, we canonically normalise the kinetic term and compare with the chiral lagrangian: Within our numerical accuracy this is the same answer as that calculated in the abelian case from the mass term which gave the value ≃ N 4π 2 . The extra massless bosons indeed seem to behave as Goldstone fields upto the commutator interactions.

Vector Mesons and Weakly Gauged Chiral Symmetries
There is one additional bosonic field in the low energy DBI action of the D7 brane -the gauge field partner of the scalar, Φ, discussed above. The field strength tensor of this gauge field enters in the standard way as 2πα ′ F ab in the square root. The leading lagrangian for this field, in a background configuration σ 0 , is where µ and ν run over the space-time indices. There is an additional term that could be added onto the DBI action. This is the Wess Zumino term which gives the coupling of the four-form C (4) to the gauge fields. We do not include this because when we calculate the equations of motion for the gauge fields, this term is only relevant for the gauge fields with a vector index on the S 3 . We will only be interested in the states that carry no SO(4) R-charge where a, b take values in the four dimensional space of the gauge theory. We write the gauge field as: The equations of motion for the gauge field are given by: = 0 (56) There is clearly a solution with M = 0 and g(ρ) =constant. This corresponds to introducing a background gauge field associated with U(1) baryon number in the field theory. Note that asymptotically in the UV the lagrangian for this field is which reflects the logarithmic running of the flavour gauge coupling. There is a second asymptotic solution of the equation of motion where g(r) ∼ 1/ρ 2 and these solutions correspond to the vector meson spectrum associated with the operatorqγ µ q. By seeking smooth solutions of the equation of motion we can determine the vector meson mass spectrum. The results of analysis are shown in the table compared to those calculated in the pure AdS background. The spectrum (of singlets on the S 3 ) in the pure AdS case is given by [21]: M 2 v R 2 = 4(n + 1)(n + 2) There are a number of interesting questions one might ask with regards the vector mesons and weakly gauged chiral symmetries. For example in models of dynamical symmetry breaking   (5) is related to the phenomenological S parameter which is well measured by the precision electroweak data. This is an UV finite quantity which in this model would be related to the difference between the coefficient of the vector F 2 term calculated above and the equivalent term for the axial gauge field. Unfortunately the DBI action has no field that corresponds to the axial gauge field. Presumably equally the operatorsqγ µ γ 5 q are described by a string mode not present in the DBI action, so it is therefore not possible for us to estimate this parameter. Similarly it would have been nice to have tested the Weinberg sum rules [35] such as, assuming vector meson dominance, where g V /A are the couplings of the lightest vector and axial vector mesons to their respective currents. We have no description of the axial vector mesons though. It would be productive to understand this sector in the future.
These issues have been highlighted in [36] where a toy model of large N QCD was proposed inspired by the general structure of the AdS/CFT Correspondence. That model has a flavour gauge field living in a (deconstructed) finite fifth dimension with symmetry breaking boundary conditions. They interpret A 5 as the pion field and the KK modes of the flavour gauge field as the vector and axial vector mesons. This is distinct from our scenario where the pion fields are described by an additional scalar field and the flavour gauge field only describes vector mesons -the two pictures do not support each other therefore.

Summary
We have analyzed chiral symmetry breaking via the gravitational dual construction of D7 brane probes in the Constable Myers non-supersymmetric geometry. The ground state of the D7 brane indeed describes chiral symmetry breaking and a vacuum manifold with massless pions in the chiral limit. We have shown that these pion fields are described by a chiral lagrangian in the infra-red as expected. We have computed the couplings of the low energy lagrangian and shown that they match expectations from naive dimensional analysis.
One might have expected significant deviations in the magnitudes of parameters because the gauge coupling of the model remains strong in the UV (as in for example a "walking" gauge theory [37]). However, the UV of the theory has an approximate N = 4 supersymmetry that forbids a chiral condensate so that above our symmetry breaking scale Λ b the condensate decreases rapidly in spite of the strong coupling. The chiral symmetry breaking behaviour of this model is therefore rather pleasingly analogous to that of QCD.