# Dynamical corrections to the anomalous holographic soft-wall model: the pomeron and the odderon

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## Abstract

In this work we use the holographic soft-wall 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 their 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 the Einstein equations in five dimensions. The results found in this work for the Regge trajectories within the anomalous soft-wall model with dynamical corrections are consistent with those present in the literature.

## Keywords

Anomalous Dimension Beta Function Regge Trajectory Full Dimension Dilaton Field## 1 Introduction

Gluons do not carry electric charges, but they have colour charge. Due to this fact, they couple to each other, as implied by Eqs. (1) and (2). The bound states of gluons predicted by QCD, but not detected so far, are called glueballs. Glueball states are characterised by \(J^{PC}\), 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.

*J*) and the square of the masses (

*m*), such that

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 \(\mathrm{SU}(N)\) for very large *N* and extended supersymmetry \((\mathcal{N} = 4)\) with a *IIB* superstring theory in a curved space, known as anti de Sitter space, or \(\mathrm{AdS}_5 \times S^5\). At low energies string theory is represented by an effective supergravity theory. Due to this, the AdS/CFT correspondence is also known as a 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 made some important progress. In these works there emerged the idea of the hard-wall model. This idea means that a hard cutoff is introduced at a certain value \(z_{\mathrm{max}}\) of the holographic coordinate *z* and the space considered is a slice of \(\mathrm{AdS}_5\) space in the region \(0 \le z \le z_{\mathrm{max}}\), 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 soft-wall model. This model introduces in the action a decreasing exponential factor of the dilatonic field that represents a soft IR cutoff. The soft-wall 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 Ref. [19] that the soft-wall 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 Refs. [20, 21, 22] was introduced the idea of using QCD beta functions to get an interesting UV behaviour for the soft-wall model modified by convenient superpotentials for the dilaton field. Then in Ref. [23] a simpler modification was proposed 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. [23] to investigate higher, even and odd, spin glueball states, and obtain the Regge trajectories related to the pomeron and to the odderon, taking into account the anomalous dimensions from some QCD beta functions with dynamical corrections, i.e., considering that the dilatonic field became dynamical satisfying the Einstein equations in five dimensions. Actually, we first consider the anomalous contributions to the original soft-wall model without dynamical corrections for higher even and odd spins, but the results are not good compared with the pomeron and odderon Regge trajectories. Then we consider the dynamical case where the results obtained are good.

This work is organised as follows: In Sect. 2 we provide a quick review of the original soft-wall model (SW) and the modifications taking into account the anomalous dimensions from QCD beta functions that we call the anomalous soft-wall model (ASW) (that is, the soft-wall with anomalous contributions as proposed in Ref. [23]). Then we introduce even and odd higher spin state operators and calculate, from the mass results, the Regge trajectories related to the pomeron and to the odderon in two different cases: the beta function with a linear asymptotic behaviour and the beta function with an IR fixed point at finite coupling. At this point, we show that the ASW model needs some corrections to provide compatible Regge trajectories related to the pomeron and to the odderon. In Sect. 3 we impose dynamical corrections coming from the dilaton field in the ASW model to achieve the Regge trajectories related to the pomeron and to the odderon. In particular, for the beta function with an IR fixed point at finite coupling, the Regge trajectories obtained for both even and odd spin glueball states, related to the pomeron and the odderon, respectively, are in agreement with those found in the literature. In Sect. 4 we make some comments and summarise our results.

## 2 The anomalous soft-wall model

Let us start this section performing a quick review of the SW model. Then we proceed to discuss the modifications due to the anomalous dimensions of the ASW model, introduce higher spin states and obtain the glueball masses and Regge trajectories.

*g*is the determinant of the metric tensor of the five-dimensional AdS space described by

*R*is called the radius of the \(\mathrm{AdS}_5\) space.

*X*, as shown in Refs. [16, 17]. The glueball masses \(m_n\), where \(m_n^2 \equiv -q^2\), can be obtained from normalisable solutions of Eq. (9):

The SYM is a conformal theory, so the beta function vanishes and the conformal dimensions have no anomalous contributions, therefore they keep only their classical dimension.

In the present work our concern is with even and odd higher spin glueballs and their corresponding Regge trajectories related to the pomeron and to the odderon.

*J*symmetrised covariant derivatives in a given operator with spin

*S*, such that the total angular momentum is now \(S+J\). In the particular case of the operator \(\mathcal{O}_4 = F^2\), one gets

*J*. Reference [27] used this approach within the hard-wall 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.

*J*, taking into account the beta function is

*J*, taking into account the beta function:

At this point, we make a brief comment on 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 renormalisation 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 [29, 30, 31]. We will also impose the requirement that the beta functions reproduce the ultraviolet perturbative behaviour analogous to QCD for small \(\lambda \) in 1-loop approximation. That is, \( \beta (\lambda ) \sim - b_0 \lambda ^2\), where \(b_0\) is a universal coefficient of the perturbative QCD beta function at leading order, given by \( b_0 = \frac{1}{8 \pi ^2}\left( \frac{11}{3} - \frac{2}{9} N_f\right) \). For a pure \(\mathrm{SU}(3)_c\) one has \(N_f = 0\), then \(b_0 = 11/24 \pi ^2\).

*z*of the \(\mathrm{AdS}_5\) space with \(\mu ^{-1}\) where \(\mu \) is defined as the renormalisation group scale. So, the relation of the beta function and

*z*is then

In the following subsections the “Schrödinger-like” equation (9) will be solved numerically for two cases, 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.

### 2.1 Beta function with a linear IR asymptotic behaviour

*W*(

*z*) is the Lambert function.

*k*, \(b_1\) and \(\lambda _0\), one can get the Regge trajectories for even and odd glueball states, which could be related to the pomeron and to the odderon, respectively.

Different values of *k*, \(b_1\) and \(\lambda _0\) used in the ASW model for the beta function with a linear IR asymptotic behaviour, Eq. (35), and the results for the Regge trajectories obtained for the pomeron and the odderon. The errors come from a linear fit

Set | | \(b_1\times 10^{3}\) | \(\lambda _0\) | Pomeron | Odderon |
---|---|---|---|---|---|

1 | \(- 0.25\) | 1.2 | 19 | \(J \approx (- 0.3 \pm 0.3) + (0.42 \pm 0.03)\, \mathrm{m}^2\) | \(J \approx (- 1.8 \pm 0.5) + (0.40 \pm 0.03)\, \mathrm{m}^2\) |

2 | \(-0.49\) | 1.2 | 19 | \(J \approx (- 0.3 \pm 0.4) + (0.39 \pm 0.03)\, \mathrm{m}^2\) | \(J \approx (- 1.3 \pm 0.4) + (0.34 \pm 0.02)\, \mathrm{m}^2\) |

3 | \(-0.72\) | 1.2 | 19 | \(J \approx (- 0.6 \pm 0.4) + (0.38 \pm 0.02)\, \mathrm{m}^2\) | \(J \approx (-1.6 \pm 0.3) + (0.33 \pm 0.01)\, \mathrm{m}^2\) |

4 | \(- 1.00\) | 1.2 | 19 | \(J \approx (- 1.0 \pm 0.3) + (0.35 \pm 0.01)\, \mathrm{m}^2\) | \(J\approx (- 2.2 \pm 0.3) + (0.32 \pm 0.01)\, \mathrm{m}^2\) |

5 | \(- 1.00\) | 1.2 | 16 | \(J \approx (- 1.7 \pm 0.2) + (0.45 \pm 0.01)\, \mathrm{m}^2\) | \(J \approx (- 3.2 \pm 0.2) + (0.43 \pm 0.01)\, \mathrm{m}^2\) |

6 | \(- 1.00\) | 1.0 | 16 | \(J \approx (- 1.7 \pm 0.2) + (0.45 \pm 0.01)\, \mathrm{m}^2\) | \(J\approx (- 3.2 \pm 0.2) + (0.42 \pm 0.01)\, \mathrm{m}^2\) |

The values for *k*, \(b_1\) and \(\lambda _0\), and the results for the Regge trajectories are presented in Table 1, where one can see that the Regge trajectories found for both the pomeron and the odderon for the beta function with a linear IR asymptotic behaviour are in disagreement with (4) for the pomeron [1], and in Eqs. (5) and (6) for the odderon [6]. For instance, in the fourth set of Table 1, one finds that the angular coefficients found for the pomeron and for the odderon are, respectively, \(0.35\pm 0.01\) and \(0.32 \pm 0.01\) GeV\(^{-2}\), which values are much higher than the expect values of 0.25 GeV\(^{-2}\) for the pomeron, Eq. (4), and 0.23 or 0.18 GeV\(^{-2}\) for the odderon, Eqs. (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.

### 2.2 Beta function with an IR fixed point at finite coupling

*W*(

*z*) is again the Lambert function and \(\lambda (z_0) = \lambda _0\) fixes the integration constant. This equation leads to the expected QCD asymptotic behaviour at short distances when

*z*is close to the boundary \((z\rightarrow 0)\):

*k*, \(\lambda _0\) and \(\lambda _{*}\), one can get Regge trajectories for even and odd glueball states, which could be related to the pomeron and to the odderon, respectively. The results obtained are presented in Table 2.

Regge trajectories obtained for both pomeron and odderon from ASW model using the beta function with an IR fixed point at finite coupling, Eq. (37), and \(\lambda _*=350\). The errors come from the choice of a linear fit

| \(\lambda _0\) | Pomeron | Odderon |
---|---|---|---|

\( - 0.36 \) | 18.5 | \(J \approx (- 0.9 \pm 0.4) + (0.51 \pm 0.03)\, \mathrm{m}^2\) | \(J \approx (- 3.0 \pm 0.9) + (0.53 \pm 0.06)\, \mathrm{m}^2\) |

\(-0.16\) | 18.5 | \(J \approx (- 4 \pm 2) + (1.4 \pm 0.3)\, \mathrm{m}^2\) | \(J \approx (- 19.7 \pm 0.3) + (3.06 \pm 0.04) \,\mathrm{m}^2\) |

\(-0.36\) | 10.5 | \(J \approx (- 1.93 \pm 0.03) + (1.34 \pm 0.01)\, \mathrm{m}^2\) | \(J \approx (- 3.82 \pm 0.05) + (1.32 \pm 0.01)\, \mathrm{m}^2\) |

\(- 0.36\) | 25.5 | \(J \approx (- 3 \pm 2) + (0.57 \pm 0.10)\, \mathrm{m}^2\) | \(J \approx (- 14 \pm 2) + (1.0 \pm 0.1)\, \mathrm{m}^2\) |

From Table 2, one can see that the Regge trajectories found for both the pomeron and the odderon for the beta function with an IR fixed point in the finite coupling case 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 0.23 or 0.18 GeV\(^{-2}\), Eqs. (5) and (6). So, we conclude that the ASW model with the beta function with an IR fixed point at finite coupling does not give reasonable results for the Regge trajectories for the pomeron or the odderon.

## 3 The dynamical corrections to the anomalous soft-wall model

In this section we will apply dynamical corrections to modify the ASW model to investigate if these corrections can provide Regge trajectories for the pomeron and the odderon compatible with those found in the literature. To do this, let us perform a quick review of the dynamical soft-wall (DSW) model, discussed in [32, 33, 34].

*D*action for the graviton–dilaton coupling in the string frame is given by

*D*action for the scalar glueball in the string frame is given by [17, 18]

### 3.1 Beta function with a linear IR asymptotic behaviour

Using the beta function given by Eq. (35) and replacing it in Eq. (59) one can solve numerically the Schrödinger-like equation for the DSW model (56) for both even and odd glueball states.

*k*, \(b_1\) and \(\lambda _0\), one can get Regge trajectories for even and odd glueball states. The set used together with the results obtained for Regge trajectories for both the pomeron and the odderon are shown in Table 3.

| \(b_1 \times 10^{3}\) | \(\lambda _0 \) | Pomeron | Odderon |
---|---|---|---|---|

\(- 0.25\) | 1.2 | 19 | \(J \approx (- 14 \pm 1) + (4.1 \pm 0.3)\, \mathrm{m}^2\) | \(J \approx (- 21 \pm 1) + (5.0 \pm 0.3)\, \mathrm{m}^2\) |

From Table 3, one can see that the Regge trajectories found for both the pomeron and the odderon with the beta function with a linear IR asymptotic behaviour, Eq. (35), are in disagreement with those found in (4) for the pomeron [1], and in Eqs. (5) and (6) for the odderon [6]. The angular coefficients are too high and the intercepts are too low, both for the pomeron and for the odderon. Other sets of parameters give even poorer results.

### 3.2 Beta function with an IR fixed point at finite coupling

Using the beta function given by Eq. (37) and substituting it into Eq. (59), one can solve numerically the Schrödinger-like equation (56) for both even and odd glueball states.

*k*, \(\lambda _0\) and \(\lambda _{*}\) and get the masses of the glueball states with even and odd spins and the Regge trajectories related to the pomeron and to the odderon, respectively. The sets of parameters used and the results obtained for the glueball state masses with even and odd spins are presented in Table 4. The results obtained for Regge trajectories are shown in Table 5.

Masses (GeV) for the glueball states \(J^{PC}\) with even and odd *J* with \(P=C=\pm 1\) calculated from the anomalous dynamical soft-wall model, Eqs. (56) and (59), and the beta function with an IR fixed point at finite coupling, (37), using five sets of parameters *k* (GeV\(^{2}\)), \(\lambda _0\) and \(\lambda _{*}\) (dimensionless)

Set | Parameters | Glueball states \(J^{PC}\) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

| \(\lambda _0\) | \(\lambda _{*}\) | \(0^{++}\) | \(2^{++} \) | \(4^{++}\) | \(6^{++}\) | \(8^{++}\) | \(10^{++}\) | \(1^{-\,-}\) | \(3^{-\,-} \) | \(5^{-\,-}\) | \(7^{-\,-}\) | \(9^{-\,-}\) | \(11^{-\,-}\) | |

1 | 0.16 | 18.5 | 350 | 1.69 | 3.28 | 4.76 | 6.23 | 7.67 | 9.12 | 4.02 | 5.50 | 6.95 | 8.40 | 9.84 | 10.00 |

2 | 0.09 | 18.5 | 350 | 1.62 | 2.84 | 4.00 | 5.14 | 6.26 | 7.37 | 3.42 | 4.57 | 5.70 | 6.82 | 7.93 | 9.04 |

3 | 0.04 | 18.5 | 350 | 1.56 | 2.52 | 3.43 | 4.32 | 5.19 | 6.05 | 2.98 | 3.88 | 4.76 | 5.62 | 6.48 | 7.32 |

4 | 0.09 | 10.5 | 350 | 0.79 | 2.13 | 3.28 | 4.39 | 5.48 | 6.57 | 2.72 | 3.84 | 4.94 | 6.03 | 7.11 | 8.19 |

5 | 0.09 | 18.5 | 250 | 1.64 | 2.86 | 4.02 | 5.16 | 6.28 | 7.39 | 3.44 | 4.59 | 5.72 | 6.84 | 7.95 | 9.05 |

Regge trajectories obtained for both pomeron and odderon from the anomalous soft-wall model with dynamical corrections, Eqs. (56) and (59), using the beta function with an IR fixed point at finite coupling, (37), for the sets of parameters presented in Table 4. The errors come from the choice of a linear fit

Set | Pomeron | Odderon |
---|---|---|

1 | \(J\approx (0.6 \pm 0.5) + (0.12 \pm 0.01)\, \mathrm{m}^2\) | \(J \approx (- 0.1 \pm 0.4) + (0.10 \pm 0.01)\, \mathrm{m}^2\) |

2 | \(J \approx (0.4 \pm 0.5) + (0.19 \pm 0.02)\, \mathrm{m}^2\) | \(J \approx (- 0.4 \pm 0.4) + (0.15 \pm 0.01)\, \mathrm{m}^2\) |

3 | \(J\approx (0.1 \pm 0.5) + (0.28 \pm 0.02)\, \mathrm{m}^2\) | \(J \approx (- 0.8 \pm 0.4) + (0.24 \pm 0.02)\, \mathrm{m}^2\) |

4 | \(J \approx (0.9 \pm 0.5) + (0.23 \pm 0.02)\, \mathrm{m}^2\) | \( J \approx (0.1 \pm 0.4) + (0.18 \pm 0.01) \mathrm{m}^2\) |

5 | \(J \approx (0.4 \pm 0.5) + (0.19 \pm 0.02)\, \mathrm{m}^2\) | \(J \approx (- 0.4 \pm 0.4) + (0.15 \pm 0.01)\, \mathrm{m}^2\) |

## 4 Discussion and conclusions

In this work we used the anomalous dimension soft-wall model, related to QCD beta functions to obtain high spin glueball masses and the corresponding Regge trajectories. Then we take into account the dynamical corrections caused by the dilaton field, such that the model becomes 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 Sect. 2, is due to the fact that although the ASW model worked well for scalar glueball states and its radial (spin 0) excitations [23], this model seems not to 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 fourth set of parameters *k*, \(\lambda _0\) and \(\lambda _*\), shown in Table 4, the values were found for the masses of the glueball states, both for higher even and odd spins, to be comparable with those found in the literature. The same beta function still provides Regge trajectories, for the pomeron and the odderon, as shown in Eqs. (60) and (61), respectively, in agreement with [1], for the pomeron, and in [6] for the odderon within the non-relativistic constituent model.

*z*), simulating a wall or in this case a hard-wall. As is well known, the hard-wall model (with Neumann boundary condition) gives good results for even [27] and odd spins [28] related to the pomeron and to the odderon, respectively. In fact, the masses found in Table 4 are pretty close to the ones obtained from the hard-wall model with Neumann boundary conditions.

So, we conclude that the dynamical corrections lead to effective potentials that work like a hard-wall 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 soft-wall model [35] and for a modified (analytical) soft-wall model [19].

## Notes

### Acknowledgments

H.B.-F. is partially supported by CNPq and E.F.C. by CNPq and FAPERJ, Brazilian agencies. D.L. is supported by the China Postdoctoral Science Foundation.

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