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Improving the Infra-red of Holographic Descriptions of QCD

A surprisingly good holographic description of QCD can be obtained from naive five dimensional gauge theory on a truncated AdS space. We seek to improve the infra-red description of QCD in such models by using a more sophisticated metric and an action derived from string theory duals of chiral symmetry breaking. Our metric is smooth into the infra-red and the chiral condensate is a prediction of the dynamics. The theory reproduces QCD meson data at the 10% level.

symmetry breaking [13], but here we will massage it to a phenomenological five dimensional holographic description of QCD.
The geometry we will use is that on the surface of a D7 brane in a non-supersymmetric dilaton flow deformation of the AdS/CFT Correspondence. We review its origin in more detail in the appendix, but let us stress its benefits now • The background gauge configuration in which the quarks live is non-supersymmetric (although not purely that of QCD) and has a running coupling.
• The mass gap is a result of the non-supersymmetric gauge configuration and the geometry relevant for quark physics is smooth at all radii or energy scales.
• The holographic dual of the quark bilinear is explicit in the string construction.
• The quark condensate is a prediction of the gauge configuration and is determined as a function of the quark mass.
These points go a considerable way towards addressing the inconsistencies of the first models. We will, however, continue to adopt the phenomenological approach with regards treating the background as describing an N=3 rather than N→ ∞ theory. In addition, the string theory construction can only realize a U(1) axial symmetry, and does not provide a holographic dual of the axial vector mesons. We include by hand appropriate fields to provide a non-abelian chiral symmetry and the axial vector states in the phenomenological spirit of [20,21].
One knows that the transverse parts of the vector vector and axial axial correlators in QCD interact differently with the chiral condensate in QCD. In the gravity dual one would expect the axial and vector gauge fields to in fact see distinct metrics. We can not incorporate this effect because the string model does not provide enough information. Nevertheless the model links the quark condensate to the dynamics and smooths the infra-red which should improve the description, at least in the vector sector, whilst doing no more harm in the axial sector than is done in [20,21]. In this paper we compute with our phenomenological model the masses and decay constants for the pion and the rho and a 1 vector mesons, and also the g ρππ coupling. We find that the model gives comparable predictions to the pure AdS models within 12% of the QCD values. We believe these results provide support for the robustness of the predictions of these holographic models. The geometry we propose returns to pure AdS space in the ultra-violet, so we do not address here the absence of asymptotic freedom in the gravity description. As we pointed out recently in [33], the gravity theory should only be used up to a UV cut off, corresponding to the scale at which QCD switches from perturbative to non-perturbative behaviour. Above that cut off the gravitational dynamics must become non-perturbative with its loop corrections completely dominating the classical results. The correct UV dynamics should be encoded at that cut off by correcting the values of higher dimension operator couplings. In principle, these can be tuned in the AdS/CFT approach to produce the holographic equivalent of a perfect lattice action.
As a small example of these ideas we consider the matching of the five dimensional gauge coupling in the UV. In [20,21] this coupling is matched to the perturbative result for the vector vector correlator in QCD. The AdS gravitational dual presumably describes a strongly coupled conformal theory in the UV and so the correlator behaviour matches the logarithmic result of the conformal but weakly coupled UV behaviour of QCD. It is surprising that the numerical coefficient of the log term can be matched though. Here we test how good that matching is by allowing the parameter to float and fitting it to data. We find such a fit induces roughly a 30% change in the coupling value, which provides a measure of non-perturbative corrections at the scale of matching to the strongly coupled regime of QCD. We leave attempts to further improve the UV of the theory for later work though.
Finally, it has recently been pointed out [31] that an appropriate change to the IR behaviour of the dilaton can correct the n scaling of the tower of excited ρ meson states. We have tested our model in this respect but find only a marginal improvement over the pure AdS case. This is a sign that, although our geometry describes a non-supersymmetric gauge configuration, it is still not a perfect description of QCD and work remains to be done on improving the geometric background.

Phenomenological Five Dimensional Models
The phenomenological approaches to describing QCD holographically are based on a 5d action of the form where The field U(x, r) = exp(iπ a (x, r)T a ) describes the pions produced by the breaking of a SU(N f ) chiral symmetry with generators T a . We assume that the background value of U is the identity so we are studying N f degenerate quarks. The non-abelian gauge fields A L and A R couple by left and right action on U. They will holographically describe the vector and axial vector mesons. The field σ is a function of r only and holographically describes the quark mass and qq expectation value. A non-zero value for this field will break the SU(N f ) L × SU(N f ) R chiral symmetry of the action down to the vector SU(N f ) V .

Pure AdS
In the simplest approaches [20,21], the dilaton, φ, is taken to be constant, so drops from the action. The background metric is AdS down to some boundary at r 0 which breaks the conformal symmetry and provides the theory with a mass gap.
Note that dilatation transformations in the field theory, which define the mass dimension of operators (for example if we scale x → e α x then a scalar field of dimension one scales as φ → e −α φ), are mapped to a symmetry of the metric with the radial direction scaling as an energy scale.
The Lagrangian for σ in these models is given by with resulting solutions σ(r) = m/r + c/r 3 . Here σ has does not transform under the field theory dilatations so m has dimension one and c dimension three. The two parameters m, c are fitted phenomenologically to the (degenerate) light quarks' mass and condensate.
The remaining parameter is g 5 , which in string theory duals is a prediction in terms of the gauge theory 'tHooft coupling g 2 Y M N. In the phenomenological approach though, this relation is abandoned and the value of g 5 is fitted to the vector current correlator extracted from QCD.
where J a µ (x) =qγ µ T a q. For QCD, the leading order contribution to In order to calculate this quantity from the five dimensional model, we appeal to the AdS/CFT Rµ (x, r)) acts as a source for the four dimensional vector current J a µ (x) in the limit r → ∞. It obeys the equation of motion We look for solutions of the form where Y 1 is a Bessel function of the second kind. Substituting the solution back into the action and differentiating twice with respect to the source V µ 0 gives the vector current correlator which (up to contact terms) yields Finally, comparing this to the perturbative QCD result (5) determines the 5d coupling as In [21,20] this model is used to calculate meson masses, decay constants and couplings coefficients with great success. We summarize these results in Table 1.
The matching in (10) is of course naive. One should match the gravitational theory to QCD only at the point where the QCD coupling becomes non-perturbative where gluonic corrections to the perturbative QCD result become important. It is therefore interesting to recompute the results of [21], but with g 5 being a free parameter of the model in order to see how accurate this matching is. On performing a global fit on all of the parameters, we found that the optimal value for g 5 is 5.19 which is 17% smaller than the result (12π 2 )/N from matching to perturbative QCD. We conclude that non-perturbative effects could have a significant effect.

The New Model
Our approach in this paper will be based around the D3/D7 brane string theory construction described in the appendix [12]. Here we will present the model as a 5d model in the spirit of (1). Starting with the string theory model's action (A10), we construct a phenomenological model by artificially extending the symmetry group from SU(N f ) R and add in the axial vector gauge field in (1). The model has the metric , and a radially changing dilaton and 5d gauge coupling with δ = 1/2, ∆ = √ 39/2. Note that we scaled all coordinates by a factor of R. The conformal symmetry breaking scale is fixed by the parameter b which will determine the scale Λ QCD . Since it is the only scale in the model we set it to one for computations. At the string theory level the value of R fixes the 5d gauge coupling, but here we will fix that phenomenologically to describe an N c = 3 theory so we have also set R = 1 and leftĝ 5 free. As r → ∞, the metric returns to AdS 5 , the factor e φ /g 2 5 goes to 1/ĝ 2 5 and we are left with exactly the pure AdS model. The radial dependence of the dilaton shows the model has a running coupling 1

Dynamical Quark Condensate
The chiral symmetry breaking quark condensate is determined dynamically in this model by the background metric which represents the background gauge configuration. The Lagrangian for the field σ(r) in this model is where the dot indicates differentiation with respect to r. The equation of motion for this field, which is complicated since σ occurs throughout the geometry, is given by where The large r form of the solutions is of the AdS form (note from the metric that σ here enters symmetrically with r and therefore is rescaled relative to (3) and has energy dimension one) where m and c are interpreted as the the quark mass and condensate respectively. We seek regular solutions that satisfyσ(0) = 0. There is a single such solution for each value of σ(0) indicating that the condensate c is determined for a fixed asymptotic value of m. The solutions are shown in Figure 1.
Note that when the dynamical function σ(r) is included in the metric for the model there is no singularity since one cannot reach r + σ = b. The model therefore extends smoothly down to r = 0. We do not need to impose a hard IR cut off and the conformal symmetry breaking is expressed through the parameter b only. 1 Formally in the string theory model of the appendix the running coupling may be determined by placing a D3 brane probe in the geometry as in [36]

Matching the 5d Coupling
The matching occurs at the boundary r → ∞, so the results are exactly the same as those for the pure AdS calculation, and we are lead to the identificationĝ 2 5 = (12π 2 )/N.

Vector Mesons
We look for solutions to the vector equation of motion (6) that are of the form V a µ (x, r) = V a µ (r) exp(iqx). In the V a r (x, r) = 0 gauge this gives the following equation of motion with We will interpret the rho mesons as normalisable modes of this equation, with the eigenvalues corresponding to the squared rho masses m 2 ρ = −q 2 . For these modes to be normalisable, we require that they vanish sufficiently rapidly as r → ∞. We must also impose the gauge invariant boundary condition ψ ′ ρ (0) = 0 to ensure the smoothness of the solution. The rho wavefunction ψ ρ (r) is then a solution to (17) for an arbitrary component of V a µ (r) subject to the boundary conditions lim r→∞ ψ ρ (r) = 0 and ψ ′ ρ (0) = 0. We solve the equation numerically to find the spectrum of rho masses.
For large N, one can write the vector current correlator as the sum over rho resonances where F ρ is the rho decay constant defined by 0|J a µ |ρ b = F ρ δ ab ǫ µ . In order to find F ρ , we proceed by finding the Green's function solution to (17). Imposing the completeness relation on the set of eigenfunctions one finds Generalising (8) we have It can be shown that, in terms of the Green's function, v(q, r ′ ) = [K 1 (r)∂ r G(q; r, r ′ )] r=∞ . From this, one finds Comparing this to (18) we can extract the rho decay constant The Axial Vector Mesons We write the axial vector field, A a µ = (A a Lµ − A a Rµ ), in the A ar (x, r) = 0 gauge, as perpendicular components plus a longitudinal component A µ a = A µ a⊥ + ∂ µ φ. The equation of motion for the perpendicular components A a µ⊥ with A i⊥ (x, r) = A(q, r) exp(iqx) is where K 1 (r) and K 2 (r) are the same as in (17), and K 3 (r) = Hf 3/2−δ/2+∆/2 h 3 r 3 (1+σ 2 ) −1/2 . The solutions represent the a 1 spin 1 axial vector meson if we let lim r→∞ ψ a 1 (r) = 0, ∂ r ψ a 1 (0) = 0.
We find the masses m 2 a 1 = −q 2 by numerically finding the eigenvalues of this equation. The decay constant F a 1 is found in the same way as (23).
The pion decay constant is similarly given by We can then extract the quark mass using the Gell-Mann-Oakes-Renner relation which must be obeyed for small quark masses (ignoring the m q dependence of the condensate) Results from these methods are displayed in table 1 and discussed in section 3.

The Pion
The pion and longitudinal axial gauge fields mix and one must look for a solution of the coupled field equations The regular solutions of these equations require one to fix two unknowns, the mass of the pion, −q 2 , and the ratio of the φ and π fields at r = 0. This is numerically hard. Instead one can use the values of m π taken from the Gell-Mann-Oakes-Renner relation and then find the ratio φ(0)/π(0) with φ ′ (0) = 0 which leads to φ(r = ∞) = π(r = ∞) = 0.

The Coupling g ρππ
To the order we are working the value of the g ρππ coupling can be read off from the expansion of |DU| 2 in the action. This is not entirely satisfactory since T rF 3 terms, which we don't include in the action, will also contribute. Nevertheless for comparison to [20] we will compute them for our best fit models below. In particular The π field is normalized so the expression in brackets in this last equation integrates to one.

Results
The results of the model are displayed in Table 1. We compute 6 QCD meson parameters for our fits (we do not include g ρππ ). Our model has two free parameters (after fixing g 5 phenomenologically as discussed above), b corresponding roughly to the strong coupling scale Λ and m corresponding to the light quark mass. The model therefore has the same number of free parameters as real QCD.
In the first model, A, we match b and m by demanding that we correctly reproduce m π and m ρ . In order to do this, we must set Λ b = 264.5 MeV and m = 2.16 MeV . This gives a prediction of 325.8 MeV for the scale of the quark condensate. The overall rms error for this model is 12.8% (Note ǫ rms = O ((δO/O) 2 /n) 1/2 with O the observable and here n = 4). For comparison we also reproduce the pure AdS fit to the same parameters found in [20]. That model has three free parameters, the value of the IR cut off, the quark mass and the quark condensate and is therefore less predictive.
In model B, we perform a global fit to all observables. This gives Λ b = 253.2(MeV ) and m = 2.24 MeV , with the characteristic scale for the quark condensate 311.9 MeV . The overall rms error for this model is 11.6%. Again we reproduce the equivalent pure AdS model fit for comparison.
For the best fit point we have also computed g ρππ = 4.81MeV using (29). This should be compared to the experimental result of 6.03 ± 0.07MeV and to the results in [20] of 4.48MeV although, as discussed above, the computation of this coupling is less robust than the other results. It is again interesting to test how well determined the 5d gauge coupling g 5 is by the phenomenological fit to the far UV expectation for Π V . For example if one fits Λ, m q and g 5 to correctly reproduce the three meson masses one finds g 5 = 4.36 which is 30% lower than the value 12π 2 /N c from perturbative QCD.

Conclusions
We have adapted a string theoretic model of chiral symmetry breaking to a phenomenological description of QCD. The model we have proposed goes some way towards addressing the inconsistencies of simple AdS slice holographic QCD models [20,21]. The background geometry of our model is non-supersymmetric, and it is the smooth variation of this geometry with the radial direction r that provides a mass gap, without the need for an artificial hard IR cut-off. In addition, the dual field to the quark mass/condensate operator is a natural part of the geometrical set-up with the value of the condensate being determined by the quark mass.
However, this is still a phenomenological approach in that we introduce extra fields and symmetries by hand into the model in order to describe the full pion and axial vector sectors.
Formally there is no geometric string interpretation for this system. We also treat the background as though it describes an N = 3 rather than an N = ∞ field theory by matching the 5d gauge coupling to QCD. We find that the predictions of this model match experimental results to within 12%. This model is a little more predictive than the pure AdS slice models since the condensate is dynamically determined by the geometry. The best fit is in fact a few percent worse than the AdS slice models but hopefully the theoretical improvements represent at least a moral victory. In any case one would naively have expected errors of order a few 100% in all of these models so the closeness to QCD across a range of holographic models supports the robustness of the approach.
A drawback of these models to date has been that the geometry returns to AdS for large r, meaning that the field theory is not asymptotically free in the UV. Incorrect physics in the UV will affect the strong coupling regime in the IR [33]. Here we investigated corrections to the matching of the 5d gauge coupling to naive perturbative QCD results. We found that this coupling's value should be changed at the 30% level indicating the size of non-perturbative effects. In the future one might hope to study the importance of higher dimension operators in the IR physics as well.

Appendix A -The String Theory Progenitor
The phenomenological model used here is based on the AdS/CFT Correspondence realization of chiral symmetry breaking in [13]. That model consists of a dilaton flow deformed AdS geometry where and the dilaton and four-form are given by There are formally two free parameters, R and b, since We can see that dimensionally b has energy dimension one and enters to the fourth power. The SO(6) symmetry of the geometry is retained at all r. We conclude that in the field theory a dimension four operator with no SO(6) charge has been switched on. b 4 therefore corresponds to a vev for the operator TrF 2 .
Quarks are introduced by including probe D7 branes into the geometry. As shown in figure   1, strings which stretch between the D3 and D7 branes are in the fundamental representation of the SU(N) gauge theory on the D3. The length of the minimum length string between the two branes determines the mass of the quark field. We minimize the D7's world-volume in the spacetime around the D3 branes. This is encoded by the Dirac Born Infeld action in Einstein frame of the D7 brane where the pull back of the metric P[g ab ] is given by Substituting from the geometry above we can find the equation show that a dynamical mass is formed for the quarks. A massless quark would correspond to a D7 brane that intersects the D3 brane so there was a zero length string between them. We see that the D3s repel the D7 and for all configurations there is a minimum length string. The solution which asymptotically has m = 0 also explicitly break the U(1) symmetry in the 8, 9 plane by bending off the axis. This is the geometric representation of the breaking of the U(1) axial symmetry of the quarks.
Letting U(r, ξ) = exp(iπ(r, ξ), this gives an action for the pion field and the σ field. There is also a superpartner U(1) gauge field in the action which describes the operatorqγ µ q and hence vector mesons. This is introduced as a gauge field F ab living on the D7 S = −τ 7 d 8 ξ e φ − det(P[g ab ] + 2πα ′ e −φ/2 F ab ) which, expanded to second order gives S = −τ 7 d 8 ξ e φ √ −g(1 +σ 2 ) Now, if we assume that the fields do not have any components on the three sphere (which is appropriate for duals to non-sypersymmetric fields) we arrive at the 5d action Between (A9) and (A10) we have rescaled U → R 4 π √ τ 7 U and redefined the metric, dilaton and g 5 to the appropriate notation for a 5d model. In particular g 5 and the dilaton now have additional r dependence which is just that found in (12). The asymptotic large r value of g 5 is given by 16π 3 g s α ′2 /r 2 . Normalizable solutions of the fields in this model in the fifth direction correspond to physical states in the gauge theory, with the quantum numbers of the operators described by the holographic field. Integrating over r then leaves the four dimensional effective Lagrangian for these states from which masses and couplings can be read off. A more complete analysis of this model can be found in [12,13,35]. This method can be extended to give a theory with an SU(N f ) V × U(1) A symmetry by replacing the single D7 brane with a stack of N f D7 branes (at large r the theory becomes supersymmetric and there is a superpotential term linking the adjoint matter fields and the quarks which breaks the SU(N f ) A symmetry). We must, however, be careful to keep N f << N so that we can still treat the stack of D7 branes as a probe, and ignore any back reaction on the geometry.