Dimension Six Corrections to the Vector Sector of AdS/QCD Model

We study the effects of dimension six terms on the predictions of the holographic model for the vector meson form factors and determine the corrections to the electric radius, the magnetic and the quadrupole moments of the rho-meson. We show that the only dimension six terms which contribute nontrivially to the vector meson form factors are X^2F^2 and F^3. It appears that the effect from the former term is equivalent to the metric deformation and can change only masses, decay constants and charge radii of vector mesons, leaving the magnetic and the quadrupole moments intact. The latter term gives different contributions to the three form factors of the vector meson and changes the values of the magnetic and the quadrupole moments. The results suggest that the addition of the higher dimension terms improves the holographic model.

Introduction. -The significant progress of the holographic duals of QCD (based on [1]) in determination of basic hadronic observables (see, e.g., Refs. [2]- [21]) suggests for further development. In this paper, we work in the vector sector of the AdS/QCD model with the hard-wall cutoff, proposed in the Ref. [2]. We study the effects of dimension six terms on the vector meson form factors and extract the values of observables such as the ρ-meson's electric radius, the mass, the decay constant, the magnetic and the quadrupole moments.
The leading order contribution to the vector meson form factors coming from the F 2 term has already been studied in detail in Refs. [3,4], where it has been shown that the holographic models in Refs. [2,5] reproduce only the trivial structure of vector mesons. In particular, instead of three independent form factors that describe vector meson, these holographic models predict only one.
We show that the inclusion of dimension six terms changes the situation towards a more interesting scenario in which all of the three form factors are corrected in different amounts. We also observe, that the only dimension six terms which give nontrivial contribution to the vector meson form factors are X 2 F 2 and F 3 . The contribution from the rest of the dimension six terms can be removed by the redefinition of the coupling constant g 2 5 . We find that the addition of a term such as X 2 F 2 is equivalent to the AdS metric deformation and, according to Ref. [6], this, in turn, is equivalent to the inclusion of the vacuum condensates. This is in agreement with the point made in Ref. [2] that the higher dimension (HD) operators which appear in the operator product expansion of QCD arise in the holographic model from the higher terms in the 5D lagrangian such as X 2 F 2 . We also notice that the term X 2 F 2 doesn't alter the values of the magnetic and the quadrupole moments, however, changes the values of the vector meson electric radius, the mass and the decay constant.
The paper is organized as follows, in Section II, we go through the basics of the holographic model given in Refs. [2,7], and in particular, we discuss the leading order action, the equations of motion for the vector bound states and the forms of dimension six terms that can enter the action. In Section III, we demonstrate that the term like X 2 F 2 doesn't change the values of the magnetic and the quadrupole moments and that its effect is equivalent to the AdS metric deformation. We also discuss, how this term, to a first approximation, changes the values of the ρ-meson mass, the decay constant and the electric charge radius. In Section IV, we consider the relevant part of the F 3 lagrangian and calculate the threepoint function which is then used in Section V to derive the corrections to the form factors of vector mesons. In Section VI, we calculate the charge radius, the magnetic and the quadrupole moments of the ρ-meson and compare these with the predictions from the other models given in Refs. [23]- [28]. Finally, we summarize the paper and also show that the form factor of pion can get corrections only from the term like X 2 F 2 .
Preliminaries. -We are working in the background of the sliced AdS metric of the form: where η µν = Diag (1, −1, −1, −1), z = z 0 imposes the IR hard wall cutoff, (with z 0 ∼ 1/Λ QCD ) and z = ǫ → 0 determines the position of UV brane. From the dictionary of the AdS/QCD model, we will correspond to the 4D vector current J a µ (x) =q(x)γ µ t a q(x) a bulk gauge field A a µ (x, z) whose boundary value is the source for J a µ (x). The 5D gauge action in the AdS 5 space is where M , (M, N = 0, 1, 2, 3, z; µ, ν = 0, 1, 2, 3 and t a = σ a /2, where σ a are usual Pauli matrices with a = 1, 2, 3). We work in the A z = 0 gauge and require ∂ µ A µ = 0.
Working in the Fourier image representation and defining A a µ (q, z) = A a µ (q)A(q, z), we can determine the linearized equation of motion for A(q, z), which is with boundary conditions A(q, 0) = 1 and ∂ z A(q, z 0 ) = 0. In general, the 5D gauge theories are not renormalizable, since the 5D gauge coupling g 2 5 has negative mass dimension. This means that these theories can only be considered as an effective theories below some scale Λ. In particular, for our case, the cutoff scale Λ should be set by 1/g 2 5 . Since, the holographic model is an effective theory with physical cutoff scale Λ ∼ 1/g 2 5 , we are free to add HD terms into the lagrangian which respect all the required symmetries. The coefficients in front of the dimension six operators are of the form c/Λ, where c is some dimensionless constant and Λ = v/g 2 5 (it can be estimated that v ∼ 24π 3 ). In general, since g 2 5 = 12π 2 /N c , according to Ref. [2], we have c/Λ = 12π 2 c/(vN c ) ∼ c/N c and, therefore, for large N c the HD terms are N c -suppressed.
There are three groups of dimension six terms one can add into the AdS/QCD lagrangian, which may contribute to the three-point function, where ∇ M is a covariant derivative, R MN P K , R MN are Riemann and Ricci curvature tensors and R is a Ricci scalar. Here, we will ignore the backreaction of the matter on the metric of the AdS space. As a result, the contribution from the terms of the second group becomes formal, since in the AdS space these terms are proportional to F 2 and can be absorbed into the coupling g 2 5 . Using the equation of motion it can be shown that the term (∇ M F MN ) 2 doesn't contribute to the two-point and three-point functions. Notice, that the terms F MN ∇ 2 F MN and (∇ A F MN ) 2 are equivalent, since they differ by a full covariant derivative which vanishes after the integration because of the boundary conditions on the fields. The terms in the third group contribute to the three-point function in such a way that the magnetic and the quadrupole moments remain unchanged. We will show this on the example with the X 2 F 2 term. The remaining dimension six terms which can contribute to the three-point function are given in the second line of the first group. Using the properties of the covariant derivatives and the equation of motion, it can be shown that where we indicated only the parts which are not expressed through the terms in the second group or through the terms which don't contribute to the three-point function. The last term enters into the action as It can be shown that this term doesn't contribute to the vector meson form factors. There are different ways to see this. One of the ways is, to notice, that the form factor is obtained as a double residue of the three-point function (see, e.g., Ref. [3]). Then, working in the Fourier image representation, we have A(q, 0) = 1 and, therefore, the term [A µ , F µν ] z=0 can't have any poles. As can be seen from the Eq. (25), only the F zν = A ν (q)∂ z A(q, z = 0) term in (6), that has poles on the UV boundary. Therefore, since, we have only one term which has poles, the double residue will vanish, leading to zero corrections for the vector meson form factors. The similar arguments are applied for the term (∇ K F MN )(∇ N F K M ). It appears, that only the term F 3 in this group that can give non zero corrections to the form factors of vector mesons.
The terms of the first group , contribute to the two-point function only through the terms in the second group. Therefore, the effect of these terms on the two point function is trivial and can be absorbed by the coupling g 2 5 . Notice, that the F 3 term is not coming from the expansion of DBI action. In this model, F 3 term is one of the possible terms which should be invariant under the Lorenz and gauge transformations. We also allow the violation of the 5D discrete charge conjugation symmetry (C) in the AdS background. As we will show, the corrections associated with this C-violating term are 1/N 2 c suppressed and the precise knowledge of either the magnetic moment or the electric charge radius of the ρ-meson may allow to determine the holographic bounds of strong C-violation (which is not observed in 4D). The effects from the X 2 F 2 term. -Consider the correction to the action (2), of the form where κ is some constant and following Ref. [2], we have where m q is the quark mass parameter and σ plays the role of the chiral condensate. We observe that the total action can be written as where the Lorentz indexes are now governed by the flat metric η MN , p(z) = 1 − κg 4 5 v 2 (z) and it is clear that, in general, the contribution from all the terms like X 2n F 2 , (n is natural number), will modify p(z) to a function P (v(z)) ≡ 1 + C 1 g 4 5 v 2 (z) + · · · + C n g 4n 5 v 2n (z), where C n are some unknown coefficients. Therefore, the inclusion of the X 2 F 2 term corresponds effectively to the deformation of the AdS metric, that is instead of the 1/z 2 factor in the metric (1), we will have p 2 (z)/z 2 . The similar arguments are applied also for the term X † F XF .
This observation allows the direct application of the result from the Ref. [3] to the present case, leaving us with the following expression for the elastic form factors: where ψ n (z) are the solutions of the equations of motion, Therefore,F nn (0) = 1 and, since, the electric G C , magnetic G M and quadrupole G Q form factors are: one can check that at Q 2 = 0, these form factors reproduce the same values for electric charge, magnetic and quadrupole moments, as in the case with κ = 0, that is in the absence of the X 2 F 2 term. This term, however, can change masses and decay constants of vector mesons. Besides, it also changes the electric radius of the ρ-meson. Notice, that the eigenvalues of the Eq. (9) may be expressed through the eigenfunctions in the following way: Up to a first order approximation, using the same eigenfunctions as in case with κ = 0, that is with M (0) n = γ 0,n /z 0 (where J 0 (γ 0,n ) = 0) but with metric perturbation p(z), we will have for the ρ-meson mass M ρ ≡ M 1 the following result: where M (0) ρ is the mass of the ρ-meson in case κ = 0, and we used the values of parameters: m q = 2.3 MeV, σ = (327 MeV) 3 , z 0 = 1/(323 MeV), taken from the Model A of Ref. [2].
The decay constant of the ρ-meson, f ρ , in terms of the eigenfunctions of the 5D equation of motion has the form as was discussed, for example, in the Ref. [4]. The solution for ψ ρ (z) ≡ ψ 1 (z) near the z = 0 is of the same form as in case κ = 0 thus, Therefore, to lowest order in κ, we will have: where f (0) ρ is the decay constant in case when κ = 0. We can also express the electric charge radius of the ρ-meson, r2 ρ C , defined as in terms of the parameter κ. In this case, using the Eqs. (8), (11) and (18), to lowest order in the coefficient κ, the electric charge radius is: where 0.53 fm 2 is the result for the electric radius obtained in Ref. [3] (again, we used parameters taken from the Model A of Ref. [2]). The similar analysis can be applied for the case of Model B in Ref. [2], for which we have: Notice, that the coefficients in front of κ, in case of Model B are almost twice as smaller than in the Model A. Also, it is straightforward to see that the contribution from the term X † F XF can be absorbed by κ. Now, since g 2 5 = 12π 2 /N c , it follows that the corrections to the observables (∼ κg 4 5 ) are 1/N 2 c suppressed. The natural constraint on the coefficient κ should come from the requirement that the corrections to the observables are small. This means that, if N c = 3, then for the first two observables in (20), we should have |κ| ≪ 0.06 and for the third one we expect to have |κ| ≪ 0.004. Therefore, we conclude, that it is natural for the coefficient κ to satisfy the condition |κ| ≪ 10 −3 .
Corrections from the F 3 term. -The action relevant for finding the corrections to the 3-point function is where α is a new dimensionless (C-violating) parameter of the theory and the Lorentz indexes are governed by the Minkowski flat metric η µν . Therefore, using the prescription of the holographic model, for the 3-point function we will have: and K 11 (p 1 , p 2 , q) = z0 0 dz z∂ z A(q, z)A(p 1 , z)∂ z A(p 2 , z) , where we used that the functions K(p 1 , p 2 , q) are symmetric under the exchange of p 1 ↔ p 2 (to understand this, see Eq. (25)), but not p 1,2 ↔ q, (q = p 2 − p 1 ) and anticipating the on-shell limit, we applied conditions: p 2 1 = p 2 2 = M 2 , (p 1 p 2 ) = M 2 −q 2 /2 and (p 2 q) = −(p 1 q) = q 2 /2, for the diagonal transitions (one can easily generalize this to non diagonal transition). Since we are dealing with the transverse components of the gauge field, to simplify the tensor structure, we applied, as in [4], the transverse projectors Π αα ′ (p 1 ) ≡ (η αα ′ − p α 1 p α ′ 1 /p 2 1 ), etc, (that allows us to add or eliminate terms proportional to p 1α or p 2β ). The solution of the (3) for timelike momentum can be written as an infinite sum: where ψ m (z) are the solutions of the (3) with b.c. ψ m (0) = ψ ′ m (z 0 ) = 0 and q 2 = M 2 m . Then, for a spacelike momentum transfer, q 2 = −Q 2 , it follows that: , and for the diagonal n ↔ n transition: where we defined new functions as with where J (Q, z) = A(Q, z) is the solution of Eq. (3). Form Factors. -Adding the corrections to the form factor coming from the F 3 term to the leading order result from the F 2 term obtained in Ref. [3] gives for the electricG C , magneticG M and quadrupoleG Q form factors the following result where see Ref. [3] for more details. In the AdS/QCD model, with α = 0 as was shown in [3], these three form factors of vector meson are expressed through the single function F nn (Q 2 ). Besides, for Q 2 = 0, the AdS/QCD model reproduce the unit electric charge e of the meson, "predict" µ ≡ G M (0) = 2 for the magnetic moment and D ≡ G Q (0)/M 2 = −1/M 2 for the quadrupole moment, which are just the canonical values for a vector particle [22]. However, for non zero value of α the situation changes towards a more realistic scenario.
Results. -One can verify that at Q 2 = 0, we have W nn Besides where J 0 (γ 0,1 ) = 0, M = γ 0,1 /z 0 is the mass of the ρmeson and we took into account that After partial integrations and using the properties of Bessel functions we will have Now, defining w ≡ W 11 12 (0) ≃ 1.261, we find (e = 1), The electric radius of the ρ-meson is , where the first term is r 2 ρ C = 0.53 fm 2 , found in Ref. [3], and the second term in the square brackets is the correction to the ρ-meson's radius. Using the Eqs. which is ≃ −0.255 fm 2 . Therefore, Now, in terms of σ, the magnetic and quadrupole moments of the ρ-meson are: µ ≃ 2 + 6.1σ and DM 2 = 1.46σ − 1. The table of possible values for electric radius, magnetic and quadrupole moments in terms of reasonable range of values for σ is given below: where r 2 ≡ r2 ρ C / fm 2 . These results depend explicitly on α (or σ) and implicitly on z 0 which is fixed by the mass of the ρ-meson. Notice, that g 4 5 |α| < 0.23, therefore, we are not outside of the perturbative domain and our calculations are consistent. For comparison with other models, see table below It is interesting, that the only HD term in the 5D effective theory that can alter the canonical values of the magnetic and the quadrupole moments is the C-violating term F 3 . Therefore, the more precise knowledge of either one of these observables (µ, D or r 2 ) can put more stringent constraints on the C-violating coefficient α. Here, we showed that the corrections are proportional to αg 4 5 and thus, are 1/N 2 c suppressed as expected. Finally, our estimates suggest that |α| < 10 −4 . Summary. -In this paper, as one of the possible ways to test and improve the AdS/QCD model proposed in the Ref. [2], we considered the addition of dimension six terms into the vector sector of the AdS/QCD lagrangian and study their effect on the vector meson form factors.
We discussed that ignoring the backreaction of the matter to the metric, the effect from the terms of the second group involving the AdS curvature tensors and Ricci scalar, is equivalent to the redefinition of the coupling g 2 5 . We showed that the term, like X 2 F 2 , doesn't change the electric charge, the magnetic and the quadrupole moments, but affects the charge radius, the masses and the decay constants of the vector mesons. The effect of this term is equivalent to the AdS metric deformation and, in agreement with [2] and [6], it is also equivalent to the addition of the vacuum condensates. However, one should keep in mind that the metric deformations are also coming from the matter fields, which we ignore compared to the explicit or effective metric deformations from the X 2 F 2 term. By calculating the form factors, we found a relation between electric charge radius, mass and decay constant of the ρ-meson on the coefficient κ (to lowest order) with which the term X 2 F 2 enters the action. Also, we expressed electric radius, magnetic and quadrupole moments of the ρ-meson in terms of the dimensionless parameter α, with which the term F 3 enters the action. These results can be straightforwardly generalized to the case of the soft wall model [4,5].
It is also interesting to study the contribution of the dimension six terms to the form factor of pion. As it was discussed in Ref. [20], in the full AdS/QCD model the pion form factor is derived from the variation of the action with respect to the two longitudinal axial-vector fields and one transverse vector field. As a result, only the term like X 2 F 2 can contribute to the form factor of pion. To demonstrate this, first, consider the term F 2 A F V , where F A is related to the axial-vector field. This term may contribute to the three-point function in such a way that only the linear pieces of the field strength tensors can enter. However, since these linear pieces vanish for the longitudinal axial-vector field, there can't be any contribution from the term like F 3 to the form factor of pion (this question was also discussed in Ref. [21]).
The other relevant dimension six terms (∇ A F MN ) 2 and (∇ K F MN )(∇ N F K M ) also can't contribute to the form factor of pion. We demonstrate this on the example with the term (∇ A F MN ) 2 which, as shown above contributes to the action in the form given in Eq. (6). However, this term contains two field strength tensors, and at least one should vanish for the longitudinal components. Similar arguments can be also applied for the second term.
Finally, we think that the results obtained here are in the range of the values from the other models. This is encouraging and suggests that the further addition of the HD terms can improve the holographic dual model of QCD.