AdS/QCD oddball masses and Odderon Regge trajectory from a twist-five operator approach

In this work, we consider a massive gauge boson field in AdS$_5$ dual to odd glueball states with twist-5 operator in 4D Minkowski spacetime. Introducing an IR cutoff we break the conformal symmetry of the boundary theory allowing us to calculate the glueball masses with odd spins using Dirichlet and Neumann boundary conditions. Then, from these masses we construct the corresponding Regge trajectories associated with the odderon. Our results are compatible with the ones in the literature.


I. INTRODUCTION
The history of what would come to be called glueballs goes back to the early days of hadronic physics, before the emergence of QCD. At that time, hadron-hadron scattering processes at high-energies and low transferred momenta, written in terms of the Mandelstan variables, s m 2 −t, were ruled by Regge theory. In that scenario, Regge proposed that in the hadronic processes "particles were exchanged", for instance, as a meson (ρ, ω, etc.) or as a "Reggeons". In both cases, their scattering amplitudes, in the t channel, behaves like A(s, t) ∼ s α(t) . If one considers a family or a set of resonances sharing the same quantum numbers, one can display them in a plane [t ≡ m 2 , α(t) ≡ J] fulfilling a linear relationship written as where J is total angular momentum, m is the mass of the Reggeized particle, α and α 0 are two constants. The above relationship plotted in a Chew-Frautschi plane is known as the Regge trajectory.
If these Reggeized particles are Reggeized gluons, one has the so-called glueballs. Glueballs are represented by their total angular momentum J and their vacuum quantum numbers P , C and I, where P is the P −parity (or spatial inversion), C is the C−parity (or charge conjugation) and I is the isospin. By using the spectroscopy notation one has J P C , omitting the isospin I since it is zero for all states considered here. For a review on glueballs, one can see Ref. [1].
Odd spin glueballs are particularly interesting because they lie on the Regge trajectory of an exchanged Reggeon called odderon.
In the context of perturbative QCD, the odderon is described by the Bartels-Kwiecinski-Praszalowicz (BKP) equation [2][3][4], as a colorless C-odd three reggeons (gluons) compound state in the t channel, as can be seen pictorially in Fig. 1. An interesting review on the odderon physics can be seen in Ref. [5].
The original proposal for the existence of the odderon in early 1970s appeared in Ref. [6], the first attempts for its measurement in Refs. [7,8], and continued through the decades of 1980 and 1990 [9,10]. Note that all these collaborations did not provide reliable experimental evidence about the existence of the odderon. Recently, the outstanding efforts done in TOTEM and D0 Collaborations, analyzing the cross sections for pp and pp, and eventually their differences, ∆σ(s) = σ pp (s) − σ pp (s) ∝ ln s, supported the existence of the odderon with 3.4σ of significance [11]. In Ref. [12] the significance was improved to 5.2σ − 5.7σ. The combination of these results may be considered sufficient to give the odderon experimentally discovered.
Motivated by this recent discovery, in the present work we are interested in odd spin glueballs J −− , with (J ≥ 1). Our aim here is to contribute with new insights and proposals to compute the oddball masses and then calculate the corresponding Regge trajectory related to the odderon. To do so, we will resort to an AdS/QCD model inspired by a duality proposed by Maldacena [13]. AdS/QCD is a suitable approach to deal with QCD phenomenology in the nonperturbative regime, where glueballs are formed. The AdS/QCD model used here is known as the hardwall model, as proposed independently in Refs. [14][15][16]. In this model, conformal symmetry is broken due to the introduction of an IR cutoff z max in the holographic coordinate z and considering a slice of the anti-de Sitter (AdS) space, given by the interval  First of all, let us emphasize the main feature of this work. As the ground state for the odd spin glueballs, 1 −− is a vector object, living at the UV boundary, we start our calculation within the holographic hardwall model by relating it to a five-dimensional massive gauge boson field defined in the AdS 5 space. This procedure, which relates operators in the four-dimensional theory to fields in the bulk of five-dimensional space, represents the accomplishment of the AdS/CFT correspondence.
The twist or twist dimension, represented by τ , is given by the conformal dimension (∆) of an operator minus its spin. In particular it will be shown that the conformal dimension of the state 1 −− is ∆ = 6, and then τ = ∆ − J = 5. In this sense we are going to refer to our model as a twist-five approach.
Note that in the Ref. [24] the authors dealt with oddballs and odderon Regge trajectories, also using the hardwall model, however relating the ground state for the odd glueballs 1 −− to a massive scalar field in the AdS 5 space. In the reference [30] the authors also started their computation, within the hardwall model, from a massive boson field in AdS side.
However, among many exotic glueball states, the authors considered only one odd glueball state, namely the state 1 −− . Now let us introduce the action for a five-dimensional massive gauge boson field A m which represents the physical vector glueball at four-dimensional boundary theory, so that: Note that vector field stress tensor is assumed as F mn = ∂ m A n − ∂ n A m and M 5 is the mass of the gauge boson field. Besides g is the determinant of the metric g mn of the AdS 5 space, given by: where z is the holographic coordinate, and L is the AdS radius. From now on we take L = 1 throughout the text and η µν with signature (−, +, +, +) is the Minkowski flat spacetime metric.
By computing δS/δA n = 0, one obtains the corresponding equations of motion: Plugging the AdS metric in the above equation and considering p = z, µ, one finds: with g mn = z 2 η mn .
In order to solve the above equation, firstly, we will use an ansatz for a plane wave with four-momentum q µ , which is propagating in the in the transverse coordinates x µ , given by: where ρ is the polarization four-vector defined in the transverse space to z coordinate, and the plane wave amplitude depends on of z coordinate, only. Note that ρ ρ = η ρλ ρ λ = 1. Following Ref. [17] we are going to consider A z = 0, and then, it implies that F zn = ∂ z A n .
Besides we choose ∂ µ A µ = 0, which implies q µ µ = η µλ q µ λ = q · = 0 ensuring that the field can be written as a plane wave. Therefore, one can get At this point, we can rewrite Eq. (5) as or using (6), one has: Defining v(z) = z ψ(z) and plugging it in above equation, so that: whose solutions are given by a linear combination of Bessel (J ν ) and Neumann (Y ν ) functions: where A ν,k and B ν,k are normalization constants, the index ν = M 2 5 + 1 and m 2 ν,k = −q 2 will be the mass squared of the odd spin glueballs at the boundary. Note that k = 1, 2, 3, · · · denote radial excitations, with k = 1 for the ground state. As we are interested in regular solutions inside the bulk, we are just considering the Bessel solution and disregarding the Neumann one. Now, by plugging Eq. (11) in Eq. (6) we can construct the complete solution for the field A ρ (z, x µ ), so that: In order to get the odd spin glueball masses we are going to impose boundary conditions, such as Dirichlet and Neumann, on the vector field A ρ (z, x µ ). Before we impose those boundary conditions, one has to resort to the AdS/CFT dictionary and learn how to relate the gauge boson bulk mass (M 5 ) and the conformal dimension (∆) of the corresponding operator (O) in the four-dimensional theory. Such a relationship is written as: where, p represents the p−form index. Here we will consider p = 1.
In particular, for the glueball ground state 1 −− , it is associated to an operator O 6 at the UV, given by [20,24,42,43]: From this operator one can infer that the scaling or conformal dimension should be ∆ = 6.
As a consequence the ground state for oddballs is associated with a twist-five operator, since the twist τ is defined as the dimension minus spin, and then, τ = ∆ − J = 5.
To construct higher spin glueball states we will follow Ref. [44] where the authors proposed to raise the total angular momentum by inserting symmetrised covariant derivatives in a given operator with spin S. After this insertion one gets with conformal dimension ∆ = 6 + and total angular momentum J = 1 + . So to obtain the states 3 −− , 5 −− , etc, we take = 2, 4, · · · . Then, all odd spin states in this formulation will have twist τ = 5.
Now, replacing ∆ = 6 + in Eq. (13), one has In this work we consider all odd spin glueball states associated with p = 1 forms.

III. RESULTS ACHIEVED
In this section we will present our results for the masses of higher odd spin glueballs as well as the Regge trajectories associated with the odderon achieved from our holographic hardwall model within a twist five operator approach, considering the usual Dirichlet and Neumann boundary conditions. In order to compare our results for oddball masses, we consider as benchmarks other results found within different approaches. Those data were extracted from the literature and are summarized in can see the largest intercept reported which is close to the unity.
In particular, two different odderon's Regge trajectories were proposed in Ref. [52], which are   obtained by using a relativistic many-body (RMB) model, and based on a nonrelativistic constituent model (NRCM).

Dirichlet boundary condition
In order to apply the Dirichlet boundary condition to compute the masses of oddballs, it requires the following condition on Eq. (12) meaning that odd glueball masses will be given by the roots of the Bessel function. From the above equation, one has where ξ ν,k is the k th zero of the Bessel function of order ν. Due to a lack of experimental/theoretical data regarding higher radial excitation states for odd glueballs, we are going to focus only in the ground state and fix k = 1. Then Eq. (20) becomes As we are interest higher odd spin glueballs, let us take a look at the Bessel function index ν, in Eq. (12). Such an index is related to the bulk mass M 5 by Now, by plugging Eq. (16) in the above equation, one gets a relationship between the Bessel function index and the glueballs' angular momentum In particular, for the state 1 −− one has = 0, ∆ = 6, p = 1 and then ν = 4. The IR cutoff z max will be fixed by using the mass of this state, m D 4,1 , as an input. At this moment, we can eliminate z max by dividing an arbitrary odd spin state by the mass of the ground odd spin state 1 −− , in Eq. (21), and get an expression to compute the masses of higher odd spin glueball states [ (even) ≥ 2], so that: Note that we will choose m D 4,1 = 3.02 GeV as an input from [54]. For this chosen input, one has z max = 2.51 GeV −1 . This value for z max was obtained from a chi-squared minimization procedure with the rms error given by for the glueball masses present in Table II with Dirichlet boundary condition. Here N is the number of measurements (glueball masses) and, N p = 1 is our only free parameter (z max ).
Note that in the original hardwall model presented in Ref. [17], the ρ-meson mass was chosen to set the scale for the other particle masses. In that case, this is quite appropriate since they were considering three different meson families all with conformal dimension operator ∆ = 3. In our case, the oddballs are characterized by ∆ = 6 + . Thus it is natural for the present model to take the mass of the oddball ground state 1 −− to fix z max . Now, we are going to consider different sets of oddball states to construct possible odderon Regge trajectories. By considering the set 1 −− , · · · , 11 −− , and taking the masses in Table   II, one can construct the following Regge trajectory associated with the odderon: Analogously, for the states 1 −− , · · · , 9 −− , one gets and for 3 −− , · · · , 11 −− , one finds It is worthwhile to mention that these Regge trajectories Eqs.

Neumann boundary condition
For Neumann boundary condition on Eq. (12), it requires And then one gets By using the following property: one has where the odd glueball mass computed in the hardwall model with Neumann boundary condition is given by Here, we will fix z max = 1.89 GeV −1 with m N 4,1 = 3.02 GeV, coming from [54], as an input. The value of z max is determined by a chi-square minimization procedure analogous to the one done for Dirichlet boundary condition but now with the masses coming from Table III. To get higher odd spin glueball states we will proceed as done for Dirichlet boundary condition, and then we can rewrite Eq. (32) as so that As before, we just consider k = 1 corresponding to nonexcited radial states. Then, from our model with Neumann boundary condition we get the set of masses, presented in Table III.   In the same way, for 1 −− , · · · , 9 −− , and for 1 −− , · · · , 5 −− , Once again, these Regge trajectories Eqs. (36)- (38) were obtained from a standard linear regression method by using the glueball masses from Table III In order to compare these results for the glueball masses, we are going to calculate the rms error with Eq. (25). Taking the values of the glueball masses from 1 −− to 7 −− of the vacuum correlator model in Ref. [54] as our benchmarks, from (25) with N = 4, one finds that δ RMS = 3.60% for the Dirichlet boundary condition from Table II   the hardwall model with twist-5 operator approach the Dirichlet boundary condition seems to work better.

IV. CONCLUSIONS
In this section, we present our last comments on our work and present some interpretations on our achieved results. Here, we have used the holographic hard wall model to compute the masses of odd spin glueball states from a twist-five operator approach as well as derive the corresponding Regge trajectories related to the Odderon with Dirichlet and Neumann boundary conditions.
As the oddball ground state 1 −− has spin 1 and the corresponding operator has conformal dimension ∆ = 6, the twist of this state is τ = 5. In this sense, the twist-five operator approach seemed appropriated to deal with the odd glueball ground state. To implement it, we started with a massive gauge boson field living in the AdS 5 related to the vector glueball at the boundary theory. The higher spin oddballs J −− = (1 + ) −− (with even ) are then represented by operators with conformal dimension ∆ + = 6 + so these states also have twist τ = 5.
Note that one can wonder if it is possible to accommodate higher even spin glueball states in our model. Note however that two possible even spin ground states 0 ++ and 2 ++ are a twist-4 or twist-2 objects. In our case we are dealing with just the twist-5 objects.
In order to compute the odd spin glueball masses, we had to introduce an IR cutoff z max by using the mass of ground state 1 −− as an input. For our purposes, our input was taken from the vacuum correlator model as in Ref. [54]. Besides, in this reference one can also find values for higher odd spin glueballs masses as well as other Refs. mentioned in Table I.
As one can see, the masses computed here for higher spin oddballs, by considering Dirichilet and Neumann boundary conditions (Tables II and III, respectively) are fully compatible with most of the models presented in Table I. It is worth to mention that mass for the state 3 −− computed in this work is also in agreement with the one obtained using a holographic QCD model as reported recently in Ref. [41]. It is worth mentioning that in this work the results coming from the Dirichlet boundary condition seems to give better glueball masses than the Neumann one, taking as benchmarks the results from Ref. [54], since the respective rms errors are 3.60% and 5.61%, as discussed at the end of the previous section.
Another point of interest in this work is to derive, from the odd spin glueball masses, the Regge trajectories associate with the odderon. By taking a look at the masses in Table II, within Dirichilet boundary condition, one can construct Regge trajectories for the odderon. the NRCM, also in Ref. [52].
Regarding to the Neumann boundary condition, from Table III,  Even though the hardwall model may be the simplest among the AdS/QCD models it provides good estimates of glueball masses despite the fact that the corresponding Regge trajectories are not intrinsically linear. Anyway, the hardwall model can provide approximate linear trajectories as the ones presented in this work compatible with other holographic and nonholographic approaches. In particular, one can note that the rms errors found here for glueballs are smaller than the corresponding ones for other hadrons as presented, for instance in Ref. [17].
To conclude, we should keep in mind that although the oddballs discussed here are still lacking direct observation, the odderon itself was discovered experimentally [11,12]. We hope that the oddball quest will come to a good end in future experiments.
study for glueballs at finite temperature, within softwall and hardwall models, respectively.