Dynamical corrections to the anomalous holographic softwall model: the pomeron and the odderon

In this work we use the holographic softwall AdS/QCD model with anomalous dimension contributions coming from two different QCD beta functions to calculate the masses of higher spin glueball states for both even and odd spins and its respective Regge trajectories, related to the pomeron and the odderon, respectively. We further investigate this model taking into account dynamical corrections due to a dilaton potential consistent with Einstein equations in 5 dimensions. The results found in this work for the Regge trajectories within the anomalous softwall model with dynamical corrections are consistent with those presented in the literature.


I. INTRODUCTION
Quantum Chromodynamics (QCD) is a non-Abelian Yang-Mills gauge theory that correctly describes the strong interactions. Its well known Lagrangian is given by: where ψ(x) represents the quark fields in the fundamental representation of the SU (3) gauge group and G a µν is the gluon field strength tensor. This tensor can be decomposed in a such way, as where A a ν are the gluon fields, with a = 1, · · · , 8, f abc are the structure constants of SU (3) group and g Y M is the coupling constant of Yang-Mills (strong) interactions.
Gluons do not carry electric charges, but they have color charge. Due to this fact, they coupled to each other, as implied by Eqs. (1) and (2). The bound states of gluons predicted by QCD, but not detect so far, are called glueballs. Glueballs states are characterised by J P C , where J is the total angular momentum, and P and C are the P −parity (spatial inversion) and the C−parity (charge conjugation) eigenvalues, respectively. QCD has many interesting characteristics and among them asymptotic freedom and confinement. The first one means that in the ultraviolet limit, i.e, for high energies or short distances, quarks and gluons practically do not interact with each other. In this case, one has a weak coupling regime and QCD can be treated perturbatively. On other hand, confinement means that in the infrared limit, i.e, for low energies or large distances, quarks and gluons are bounded to each other strongly, in a strong coupling regime, and therefore, inaccessible to the perturbative approach. Calculations involving bound states as the glueball masses, and consequently, its related Regge trajectories, are features of the non-perturbative regime. Regge trajectories are well known approximate linear relations between total angular momenta (J) and the square of the masses (m), such that: with α 0 and α constants.
In this paper we will calculate the masses of the even and odd spin glueball states and from this masses one gets the Regge trajectories for the pomeron and odderon, respectively.
As mentioned before those glueball states were not detected experimentally so far. Due to this, the glueball state masses come from some particular models. In the case of the pomeron, related to the even spin glueball state masses, there are some models to describe it, such as, the soft pomeron [1], given by: where the masses throughout this work are expressed in GeV. There are also other models like the triple pole pomeron [2] , the hard pomeron [3], the BFKL pomeron [3] and etc. All of these models provide trajectories pretty close to eq. (4), so we are going to compare our results obtained for the Regge trajectory of the pomeron with this one.
In the case of odderon, related to odd spin glueball states, there are also many models to describe it, such as, isotropic lattice [4], anisotropic lattice [5], relativistic many body [6], given by: the non-relativistic constituent [6] given by: and etc. We are going to compare our results obtained for the Regge trajectory of the odderon, with these two trajectories.
The AdS/CFT or Anti de Sitter/Conformal Field Theory correspondence [7][8][9][10][11] arises as a powerful tool to tackle non-perturbative Yang-Mills theories. The AdS/CFT correspondence relates a conformal Yang-Mills theory with the symmetry group SU (N ) for very large N and extended supersymmetry (N = 4) with a IIB superstring theory in a curved space, known as anti de Sitter space, or AdS 5 × S 5 . At low energies string theory is represented by an effective supergravity theory, due to this reason the AdS/CFT correspondence is also known as gauge/gravity duality.
After a suitable breaking of the conformal symmetry one can build phenomenological models that describe (large N ) QCD approximately. These models are known as AdS/QCD models.
In order to deal with conformal symmetry breaking the works [12][13][14][15] have done some important progress with this issue. In these two last works, emerges the idea of the hardwall model. This idea means that a hard cutoff is introduced at a certain value z max of the holographic coordinate, z, and is considered a slice of AdS 5 space in the region 0 ≤ z ≤ z max , with some appropriate boundary conditions.
Another holographic approach to break the conformal invariance in the boundary field theory, and make it an effective theory for large N QCD, is called the softwall model. This model introduces in the action a decreasing exponential factor of the dilatonic field that represents a soft IR cutoff. The softwall model (SW) was proposed in [16] to study vector mesons, and subsequently extended to glueballs [17] and to other mesons and baryons [18].
It was shown in the ref. [19] that the softwall model does not give the expected masses for the scalar glueball states (and its radial excitations) and higher spin glueball states (with even and odd spins).
In ref. [20], it was introduced the idea of using QCD beta functions to get an interesting UV behaviour for the softwall model modified by convenient superpotentials for the dilaton field. Then, in reference [21] it was proposed a simpler modification of the SW model, taking into account anomalous dimensions, also related to QCD beta functions to obtain the scalar glueball spectra and its (spin 0) radial excitations. The resulting masses found for some choices of beta functions are in agreement with those presented in the literature.
In this work, our main objective is to extend the previous studies done in ref. [ where Φ(z) = kz 2 is the dilatonic field, M 5 is the 5-dimensional mass of the scalar field X = X(z, x), and g is the determinant of the metric tensor of the 5−dimensional space AdS space described by: with M, N = 0, 1, 2, 3, 4 ; µ, ν = 0, 1, 2, 3 and η µν = diag (−1, 1, 1, 1) is the metric of the four-dimensional Minkowski space. The constant R is called the radius of the AdS 5 space.
In the SW model, to find the masses of glueball states one has to solve a "Schrödingerlike" equation: corresponding to the equation of motion for the field X, as shown in refs. [16,17]. The glueball masses m n , where m 2 n ≡ −q 2 , can be obtained from normalisable solutions of Eq.(9) as where n = 0, 1, 2, ... denote the radial (spin 0) modes.
From the AdS/CFT dictionary one can show how to relate masses of supergravity fields in the AdS 5 × S 5 space (here and in the following we are disregarding the S 5 space since the fields are assumed to be independent of it) and scaling dimensions of the local gaugeinvariant dual operator in the Super Yang-Mills theory (SYM). For instance, the classical (non-anomalous) conformal dimension ∆ class. of a super Yang-Mills scalar operator is given by: Therefore, one can write: For a pure SYM theory defined on the boundary of the AdS 5 space, one has that the scalar glueball state 0 ++ is represented by the operator O 4 , given by: The lightest scalar glueball 0 ++ is dual to the fields with zero mass (M 2 5 = 0) in the AdS 5 space, then one can see, using (12), that the operator O 4 has conformal dimension ∆ class. = 4.
The SYM is a conformal theory, so the beta function vanishes and the conformal dimensions has no anomalous contributions, therefore they keep only their classical dimension.
On the other side, for a QCD scalar glueball operator, its full dimension can be computed from the trace anomaly of the energy-momentum tensor [22,23]: where the beta function is defined as: and µ is a renormalisation scale, α ≡ g 2 Y M /4π and g Y M is the Yang-Mills coupling constant. The fermionic part in (14) can be disregarded because only the operator T rF 2 is relevant for our purposes. Moreover, the scaling behaviour for a generic operator is given by: 6 The full dimension ∆ O also can be seen as a sum of the classical dimension ∆ class. and the anomalous dimension γ(µ), so that: For the scalar glueball operator, one can insert eq. (14), without fermionic part, in (16), obtaining: or where the prime represents the derivative with respect to α.
The trace T µ µ scales classically, that means ∆ T µ µ (Class.) = 4 [23]. So, the scalar glueball operator T rF 2 has the full dimension: Using the 't Hooft coupling and now the prime represents the derivative with respect to λ and the beta function is given by: In the reference [21] this approach was used to study the scalar glueball and its radial (spin 0) excitations spectroscopy due to contributions of anomalous dimensions.
In this present work our concern is about higher spin glueballs, for even and odd spins, and to get their corresponding Regge trajectories related to the pomeron and to the odderon.
To get the Regge trajectory for even glueball states we will use the method described in [24] to raise the spin of the glueball, that is, we will insert symmetrised covariant derivatives in a given operator with spin S in order to raise the total angular momentum, such that, the total angular momentum after the insertion is now S + J. In the particular case of the operator O 4 = F 2 , one gets: with conformal dimension ∆ class. = 4 + J and spin J. The reference [25] used this approach within the hardwall model to calculate the masses of glueball states 0 ++ , 2 ++ , 4 ++ , etc and to obtain the Regge trajectory for the pomeron in agreement with those found in the literature.
So, for even spin glueball states using the softwall model after the insertion of symmetrised covariant derivatives, and using that ∆ class. = 4 + J and ∆ class. = 2 + 4 + R 2 M 2 5 , one has: In a similar way, one can write the full dimension ∆ even J T µ µ = 4 + J, and now the eq. (21) can be written as: Using (12), the full dimension for a glueball state with higher even spin J, taking into account the beta function is: or explicitly: On the other hand, for odd spin glueballs, the operator O 6 that describes the glueball state 1 −− is given by: and after the insertion of symmetrised covariant derivatives one gets: with conformal dimension ∆ class. = 6 + J and spin 1 + J. The reference [26]  For the case of the odd spin glueball states, using ∆ = J + 6 and ∆ = 2 + 4 + R 2 M 2 5 , one has: In a similar way, one can write the full dimension ∆ odd J T µ µ = 6 + J , and now the eq. (21) can be written as: Using (12), one can write the full dimension for a glueball state with higher odd spin J, taking into account the beta function: and explicitly: Then, replacing eqs. (27) and (33) in the Schrödinger-like equation (9) one can solve it numerically and get the masses of higher spin glueballs (even and odd) and consequently obtain the Regge trajectories for the pomeron and the odderon, as we will show below.
At this point, we make a brief comment about QCD beta functions. From perturbative QCD it is well known that one can express the beta function through a power series of the coupling where each term comes from a certain loop order. Exceptionally the two first terms do not depend on the renomalisation set up, but other ones, i.e, higher order terms do.
For our purposes, we will consider some effective non-perturbative beta functions that could reproduce the IR behaviour of QCD as one can see in the works [27][28][29]. Another requirement is that the beta functions reproduce the ultraviolet perturbative behaviour analogous to the QCD for small λ in 1− loop approximation. That is: where b 0 is a universal coefficient of the perturbative QCD beta function at leading order, where the integration constant will be fixed by λ(z) ≡ λ 0 at a particular energy scale z 0 .
In the following subsections the "Schrödinger-like" equation (9) will be solved numerically for two regimes, namely, the beta function with a linear asymptotic behaviour and the beta function with an IR fixed point at finite coupling. Then, we obtain the corresponding Regge trajectories trying to fit the pomeron and the odderon for each beta function.

A. Beta function with a linear IR asymptotic behaviour
We begin considering the beta function given in [27,28], such that: This beta function behaves as −b 0 λ 2 in the UV and as −(b 0 /b 1 )λ in the IR. Solving eq. (34) for this beta function, one gets exactly: where W (z) is the Lambert function.
Substituting eqs. (35) and (36) in eqs. (27) and (33), solving numerically the Schrödingerlike equation (9) and using some sets of values for k, b 1 and λ 0 , one can get the Regge trajectories for even and odd glueball states, that could be related to the pomeron and to the odderon, respectively.  5) and (6). Since for the other sets the values for the angular coefficients are even higher, we conclude that the ASW model with the beta function with linear asymptotic behaviour does not give good results for the Regge trajectories for the pomeron or for the odderon.

B. Beta function with an IR fixed point at finite coupling
Here, we consider the beta function given in [29]: This the beta function has the necessary IR and UV requirements. This means that for the IR fixed point λ = λ * this beta function vanishes. Moreover, it reproduces the perturbative β(λ) ∼ −b 0 λ 2 at 1− loop order in the ultraviolet and behaves as β(λ) ∼ +λ 3 at large coupling. The equation (34) can also be solved exactly for this beta function, so that: where W (z) is again the Lambert function and λ(z 0 ) = λ 0 fixes the integration constant.
This equation leads to the expected QCD asymptotic behaviour at short distances when z is close to the boundary (z → 0): Replacing (37) and (38) in eqs. (27) and (33), solving numerically the Schrödinger-like equation (9) and using some sets of values for k, λ 0 and λ * , one can get Regge trajectories for even and odd glueball states, that could be related to the pomeron and odderon, respectively.
The results obtained are presented in table II.
From the table II one can see that the Regge trajectories found for both the pomeron and the odderon for the beta function with an IR fixed point at finite coupling regime are in disagreement with those found in [1], for the pomeron, and in [6] for the odderon. As in the case of the previous beta function, for instance, the angular coefficients found here are too high when compared with the ones from the pomeron 0.25, eq. (4), and the odderon   [30][31][32].
The 5D action for the graviton-dilaton coupling in the string frame is given by: where G 5 is the Newton's constant in five dimensions, g s is the metric tensor in the 5−dimensional space, Φ = Φ(z) is the dilaton field and V G is the dilatonic potential. All of these parameters are defined in the string frame. The metric g s has the following form: with M, N = 0, 1, 2, 3, 4; µ, ν = 0, 1, 2, 3, and η µν = diag (−1, 1, 1, 1) is the metric of the four-dimensional Minkowski space. Performing a Weyl rescaling, from the string frame to the Einstein frame, one can write eq. (40) as: where Varying the action (42), one can obtain the equations of motion, which are given by: where E M N is the Einstein tensor.
Using the metric parametrisation given by (41), the equations of the motion (44) and (45) can be written as: where we defined: Solving the equations (46) and (47) for the quadratic dilaton background, Φ(z) = kz 2 , one finds: and where 0 F 1 (a, z) is a confluent hypergeometric function. Using (48) and (49), one can easily see that: which implies that the metric (41) in this dynamical model is no longer AdS, but it is asymptotically AdS in the limit z → 0.
The 5D action for the scalar glueball in the string frame is given by [17,18]: and its equations of motion are: Using the metric (41), one gets: Doing the substitution B(z) = Φ(z) − 3A s (z), using the ansatz G(z, x µ ) = v(z)e iqµx µ , and defining v(z) = ψ(z)e B(z) 2 , one gets: where A is given by 0 F 1 ( 5 4 , Φ 2 9 ), or explicitly, for the quadratic dilaton Φ(z) = kz 2 : This equation was solved numerically in [32] and the masses found for the scalar glueball and its radial (spin 0) excitations are compatible with those obtained by lattice QCD.
To take into account dynamical corrections plus the anomalous dimension effects, one must recall, from section 2, that for even spin glueball states the full dimension is given by eq.(25), so that: For the odd spin glueball state the full dimension is given by eq.(31), then Using once again the relation R 2 M 2 5 = ∆(∆ − 4), one gets: To consider even or odd spin glueball states one can replace the first or the second line of eq. (59), respectively, into the Schrödinger-like equation obtained from the DSW softwall model, eq.(56), and solve it numerically. For our purposes, the same two regimes of the previous section will be studied, namely, the beta function with a linear asymptotic behaviour and the beta function with an IR fixed point at finite coupling. The results will be discussed in the following.     finite coupling regime are in agreement with those found in (4) for the pomeron [1], and with the non-relativistic constituent model, eq. (6), for the odderon [6]. In particular, the Regge trajectory related to the pomeron from table V with set 4, is given by: and is shown in the left panel of figure 1. The Regge trajectory related to the odderon from which is shown in the right panel of figure 1. Here, we excluded the state 11 −− because the corresponding Regge trajectory with it was not good when compared with the odderon results, eqs. (5) and (6). Note that this happens due to the non-linear behaviour of the quadratic masses of the odd spin glueballs which can be computed by using the values for the masses found in table IV. Note also that in the original softwall model the Regge trajectories are linear, but here in the dynamical version of this model the trajectories are no longer linear. As we are going to show on the next section, this happens because the dynamical corrections produce effective potentials which turns this model similar to the hardwall model which provides non-linear trajectories. These non-linear behaviour also lead to large uncertainties for the intercepts. As mentioned before, those uncertainties come from the usual linear (least squares) regression fit.

IV. DISCUSSION AND CONCLUSIONS
In this work we used the anomalous dimension softwall model, related to QCD beta functions to obtain high spins glueball masses and the corresponding Regge trajectories.
Then, we take into account the dynamical corrections caused by the dilaton field, such that the model become a solution of Einstein's equations in five dimensions. We take this anomalous and dynamical model to calculate the masses of glueball states with even and odd spins.
Our motivation to consider these dynamical corrections, as was shown in the section II, is due to the fact that although the ASW model worked well for scalar glueball states and its radial (spin 0) excitations [21], this model seems to does not work well for higher spin glueball states for both QCD beta functions studied, namely, the beta function with an IR fixed point at finite coupling and the beta function with a linear asymptotic behaviour.
In this work, in particular, for beta function with an IR fixed point at finite coupling, using the 4th set of parameters k, λ 0 and λ * , shown in the table V, were found values for the masses of the glueball states, both for higher even and odd spins, comparable with those found in the literature. This same beta function still provides Regge trajectories, for both pomeron and odderon, as shown in the equations (60) and (61), respectively, in agreement with [1], for the pomeron, and in [6] for the odderon within the non-relativistic constituent model.
This leaves us with a question: why the softwall models with or without anomalous corrections do not work for high spin glueballs but the model with dynamical corrections does? To answer this question we plot the effective potentials of the anomalous dynamical softwall model for high spins discussed in this work in Figure 2. The effective potentials are given by: V even J = k 2 z 2 + 15 4z 2 + 2k + 1 V odd J = k 2 z 2 + 15 4z 2 + 2k + 1 z 2 6 + J + β (λ) − 2 λ β(λ) 2 + J + β (λ) − 2 λ β(λ) . (63) One can see in Figure 2 that the effectives potentials show an abrupt rise in the IR region (large z), simulating a wall or in this case a hardwall. As is well known, the hardwall model (with Neumann boundary condition) gives good results for even [25] and odd spins [26] related to the pomeron and to the odderon, respectively. So, we conclude that the dynamical corrections lead to effective potentials that work like a hardwall at some finite value of the holographic coordinate z implying good results for the glueball masses and Regge trajectories. Similar results have also been recently found for the (non-anomalous) dynamical softwall model [33] and for a modified (analytical) softwall model [19].