Hybrid spectroscopy within the Graviton Soft-Wall model

In this analysis, the so-called holographic graviton soft-wall model (GSW), first developed to investigate the glueball spectrum, has been adopted to predict the masses of hybrids with different quantum numbers. Results have been compared with other models and lattice calculations. We have extended the GSW model by introducing two modifications based on anomalous dimensions in order to improve our agreement with other calculations and to remove the initial degeneracy not accounted for by lattice predictions. These modifications do not involve new parameters. The next step has been to identify which of our calculated states agree with the PDG data, leading to experimental hybrids. The procedure has been extended to include hybrids made of heavy quarks by incorporating the quark masses into the model.


I. INTRODUCTION
In the last few years, hadronic models, inspired by the holographic conjecture [1,2], have been vastly used and developed in order to investigate non-perturbative features of glueballs and mesons, thus trying to grasp fundamental features of QCD [3,4].Recently we have used the so called AdS/QCD models to study the scalar glueball spectrum [5,6].The holographic principle relies in a correspondence between a five dimensional classical theory with an AdS metric and a supersymmetric conformal quantum field theory with N C → ∞.This theory, different from QCD, is taken as a starting point to construct a 5 dimensional holographic dual of it.This is the so called bottom-up approach [7][8][9][10].In this scenario, models are constructed by modifying the five dimensional classical AdS theory with the aim of resembling QCD as much as possible.The main differences characterizing these models are related to the strategy used to break conformal invariance.Moreover, it must be noted that the relation which these models establish with QCD is at the level of the leading order in the number of colours expansion, and thus the mesonic and glueball spectrum and their decay properties are ideal observables to be studied by these models.The starting point for the present investigation is the holographic Soft-Wall (SW) model scheme, were a dilaton field is introduced to softly break conformal invariance.Within this scheme we have recently introduced the graviton soft-wall model (GSW) [6,11,12] which has been able to reproduce, not only the scalar meson spectrum, but also the lattice QCD scalar glueball masses [13][14][15], that was not described by the traditional SW models.Moreover, a formalism to study the glueball-meson mixing conditions has been developed and some predictions, regarding the observably of pure glueball states, has been provided [11,12].The success of the model in reproducing the scalar QCD spectra, has motivated us to extend the GSW model to describe the spectrum of the ρ vector meson, the a 1 axial vector meson, the pseudo-scalar meson spectra and high spin glueballs [16].
For forty years of the study of Quantum Chromodynamics (QCD) has served to establish one of the pillars of the Standard Model.Although gluons are now firmly established as the carriers of the strong force, their nonperturbative behavior remains enigmatic.This unfortunate circumstance is chiefly due to two features of QCD: the theory is notoriously difficult to work with in the nonperturbative regime, and experimental manifestations of glue tend to be hidden in the spectrum and dynamics of the conventional hadrons.In particular, experimental manifestations of hadrons that carry valence quark and gluonic degrees of freedom have been postulated since the early days of QCD.These states are called hybrids, and our lies in extending our previous experience with conventional hadrons to these states using the very successful GSW model [16].
Let us describe briefly the contents of this work.In Section II we summarize the essence of the GSW model [16].In Section III we apply the GSW model to calculate the spectrum of hybrid states.In Section IV we compare our results with lattice QCD and model calculations.In Section V, we present two possible modifications of the GSW model which do not involve new free parameters and allow to reproduce the essential outcome of lattice data.In Section VI we compare our results with experimental states appearing in the PDG compilation with the same quantum numbers.From Section VII to Section IX, we repeat the analysis for the heavy particles, i.e., we discuss the GSW model predictions for the spectra, we compare our results with lattice QCD, model calculations and experimental data.We end by collecting some conclusions of our study.

II. DESCRIPTION OF HADRONS IN THE GSW MODEL
In this section, the essential features of the GSW model are introduced.The development of this approach has been motivated by the impossibility of the conventional SW models to describe the glueball and meson spectra with the same energy scale [6,11,12].The main differences, which distinguishes the GSW model from the traditional SW, is a deformation of the AdS metric in 5 dimensions. where The quantities evaluated within this new metric are displayed with overline.The function φ 0 (z) will be specified later.This kind of modification has been adopted in many studies of the properties of mesons and glueballs within AdS/QCD [17][18][19][20][21][22][23][24][25][26].The metric tensor and its determinant of this new space can be related to the usual AdS 5 metric and its determinant: Once the gravitational background has been defined by the model, the same strategy used in the SW case is considered in order to obtain the equations of motion (EOMs) for the different fields dual to given hadronic states.The new action, written in terms of the standard AdS metric of the SW model, is given by where here the prefactor exp φ 0 (z) 5  2 α + β + 1 takes into account the dilaton term, as in the SW model, and also the modification of the metric.The parameters α and β parametrise the internal dynamics of the hadrons of QCD described within this holographic framework.In the AdS dynamics, α characterises the modification of the metric, while β characterises the SW model dilaton, namely the breaking of conformal invariance.Since the GSW model has been developed as a modification of the SW model, we propose to fix β to reproduce the kinetic term of the standard SW model action [6,11,12,16].Thus in the case of scalar fields, β = β s = 1 + 3 2 α and in the case of the vector fields β = β ρ = 1 + 1 2 α.The function L(x µ , z) is the Lagrangian density describing the motion of the dual fields in the space described by the metric Eq. ( 1).The dilaton profile function φ 0 adopted in the GSW model is the same of the one usually addressed in SW based models [6,17,18,24,[27][28][29][30], i.e.φ 0 (z) = k 2 z 2 .The action characterizing the fields propagating in the AdS 5 space contains a mass like term whose value is fixed as follows: where ∆ is the conformal dimension of the fields and p depends on its p-form.In other analyses, ∆ has been corrected by including the contribution of an the anomalous conformal dimension that characterises the chiral symmetry breaking mechanism [31].In particular, ∆ p = 0 for scalar, vector and tensor mesons, and ∆ p = −1 for pseudoscalar and axial vector mesons.Here and in the next sections, we the GSW predictions for the hybrid masses without taking into account further modifications of the model.Within these scheme, hybrids and multiquark states, defined as quark and gluon operators in QCD, will be described by the properties of their p-forms.For example, a vector field can be described as [16] V µ = Ψγ µ Ψ (6) thus p = 1, ∆ = 3 and therefore M 2 5 R 2 = 0. Within the present prescription, an an axial vector field has M 2 5 R 2 = 0.In the next section, we extend the calculation of the AdS 5 mass to hybrids.

III. DESCRIPTION OF THE LIGHT HYBRIDS AND THEIR SPECTRUM
Hybrids are hadrons formed of valence quarks and valence gluons.Due to the non perturbative nature of this bound states, the use of models is needed to predict and describe possible properties of these systems, see e.g.Refs.[32][33][34][35][36][37].In this scenario, the GSW model, already successfully applied to the studies of glueballs and regular mesons, can be used to calculate the spectra of hybrids with different quantum numbers.In particular, let us start with only non strange light hybrids described as quark-antiquark color octet coupled to a valence gluon leading to a hadronic color singlet.
Let us describe the hybrid fields with the lowest conformal dimensions and characterized by the following quantum numbers J P C , i.e. spin, parity and charge conjugation.Since these hybrid fields have to be gauge invariant, we will use only gauge invariant quantities for the quarks and gluon counterparts.In order to construct the fields, we will use for the quarks conventional bilinears and for the gluons their color magnetic field B a i = − 1 2 ε ijk F a ij , where F a µν is the color gauge tensor, µ, ν = 0, 1, 2, 3 are Lorentz indices, i, j, k = 1, 2, 3 are the spatial Lorentz indices and a is the color index, and the color electric field E a i = F a 0i .To describe the quantum numbers of the hybrids, we recall the transformation properties of the quark bilinears and color magnetic and electric fields under C and P , By using these properties, one can build hybrid field configurations for specific quantum numbers.A crucial role in the solution of the dual field equation of motion is played by the AdS 5 mass Eq.( 5), which strongly depends on the value of the p-forms.In order to calculate the hybrid ground states, we consider the configurations corresponding to the minimum AdS 5 mass.Taking these arguments into account, the hybrid field configurations with the lowest AdS 5 masses can be divided in two classes.The first one corresponds to those which have mesonic quantum numbers, and the corresponding hadrons are less wishful phenomenologically.These fields, to lowest order in conformal dimensions, are The second class are those field configurations with similar properties but with exotic quantum numbers, i.e. quantum numbers that cannot be obtained by mesonic quark-antiquark states.To lowest order in conformal dimensions, these fields are In the present analysis, we only study J = 0 and J = 1 light hybrids.The equation of motion for the dual fields of these states are essentially those corresponding to scalar and vector fields, respectively.As shown in Ref. [16], the potential term strongly depends on the AdS 5 masses given in Eqs. ( 9) and (10).Let us point out that the EOMs can be re-arranged as a Schrödinger like equation.For the scalars, the EOMs reads: and the EOMs for the vectors are where M 2 is related to the mode energies.We must recall that the parameters of the model have been fixed in our previous works, being α = 0.55 ± 0.04 and k = 370 √ α MeV [16].For the sake of simplicity, we display results corresponding to α = 0.55.11) and (12) with the values of the conformal masses shown in Eqs. ( 9)- (10).
Eqs. ( 11) and ( 12) have been numerically solved and the corresponding results are displayed in Table I.Since the GWS is supposed to be a faithful 1/N C leading order representation of QCD, we cannot consider e.g., the gluon a mass as addressed for other conventional model calculations.
The GSW model calculations leads to values of the hybrid masses within the range (2.07-2.15GeV), similar to other model calculations [32-37, 43, 44].However, as one can notice, the masses of the lightest particles differ between the predictions of mentioned collaborations.For the much sought exotic 1 −+ , the corresponding mass, obtained from the GSW model, is similar to that of the three LQCD calculations shown [38][39][40] but too high if compared with the Bethe-Salpeter approach [43,44].We also agree quite well with the LQCD [39] and LQCD(renormalon) calculation of Ref. [40] in three states.However, our 0 +− and 0 −− appear to be very high compared to the Bethe-Salpeter approach [43,44], but reasonable compared with the Constitutent Gluon [36,37] and low compared to LQCD [39].To summarize, we report that the 0 −+ state is well reproduced while, the other states are underestimated except Hybrid masses as in ref. [45] Bag [32,33] Flux Tube [34,35] Constituent Gluon [36,37] LQCD (mπ = 396 MeV) [39] GSW 0 −+ 1.3 1.7-1.99)-( 10) and in refs.[32][33][34][35][36][37][38][39][41][42][43][44][45][46].The theoretical errors in the GSW predictions are due to the uncertainty on the α parameter.Masses given in GeV unity.the 1 −+ case.Remarkably, these calculations, which do not involve any free parameter, are in line with those of other models or LQCD.In particular, we predict that scalar hybrids are lighter than the vector ones, as addressed by lattice data, except for the 1 −+ .However, as one might notice, the Lattice calculations clear indicate that only few states can be degenerate, for example, the mass of the 0 −+ is lower than that of the 0 ++ .In order to remove this degeneracy, in the next section we propose different possible modifications of the GSW model.Let us remark that, in the next section, only some possible simple extensions of the GSW model will be discussed in order to highlight how the present approach can be improved without the inclusion of further free parameters.Such a choice is essential to preserve the relevant predict power of the present model.It is also worth stressing that the holographic approach represents a 1/N c leading order calculation, which is good to determine hybrids with masses around 2 GeV.

V. BEYOND THE GSW MODEL
Our aim here is to propose an extended scenario to reproduce the hierarchy of the hybrid masses predicted by the lattice QCD and model calculations.The improvement that we present is based on approaches already discussed in several investigations involving holographic models [47][48][49].All these scenarios do not require any additional free parameter.As one might notice, since the present experimental and lattice scenarios are not well constrained, the main purpose of this section is to show that the GSW can be considered as a solid baseline for any future calculation or comparisons.In fact, it has been proved that the present approach is able to reproduce the almost linear trajectory of the glueball masses [6,11,12], the spectra of light and heavy mesons (scalars, pseudo-scalars, vectors and axial vectors) [12,16] with only two parameters.Let us also point out that the spirit of the modifications we are proposing is to parametrize the possible peculiar dynamics underlying the hybrid structure.In fact, we propose to keep the same wrap factor in the metric in Eq. ( 1), the profile function of the dilaton Eq. ( 4) and the parameters characterzing the model.Therefore, in order to describe hadrons with different dynamics from that of glueballs and regular mesons, some adjustment are needed.In particular, to improve the model we consider that the field interpolators might lead to anomalous dimensions that might affect the conformal mass [47][48][49]: However, there is no direct correspondence between the anomalous dimensions in QCD and the corresponding holomorphic anomalous dimensions.Let us discuss in what follows two modifications of the GSW associated with proposal ∆ p .

A. First modification
The first modification consists in introducing the anomalous dimension ∆ p that leads to distinguish scalar and vector fields from the pseudoscalar and the axialvector ones.Such a strategy was quite successful in the study of regular mesons.In this case, one assigns ∆ p = −1 for states whose field operator definition involves the γ 5 matrix [47].Using this input and Eq (13) one obtains the following conformal masses for each field interpolator of the mesonic type, For the non-mesonic type, we get: The resulting spectrum is displayed in Tab.III in the columns labeled GSWm1.As one can see, now the degeneracy between the first scalars (0 −+ and 0 ++ ) and the second ones (0 +− and 0 −− ) is removed, as that for the two vectors 1 −− and 1 +− compared to 1 ++ and 1 −+ .However, in this case, except for the 1 −+ we predict vector hybrids lighter than the scalar ones.Moreover, the masses of the 1 −− and the 1 +− states largely underestimate the lattice predictions.We conclude that this simple modification based on the phenomenology of regular mesons does not lead to a significant improvement to the GSW model.On the contrary, the hierarchy between the scalar and vector mesons is not reproduced.

B. Second modification
By following the line of the procedure described in the previous section, but trying to take into account differences between regular and hybrid mesons, we propose to introduce the anomalous dimension to eliminate the degeneracy between states with different parity, as proposed in Refs.[47,48].However, we assume that the lowest states, the 0 −+ and the 1 −− correspond to ∆ p = 0, while, those with opposite parity are associated to ∆ p = 1.This strategy is almost equivalent to that discussed in Ref. [47,48], where there is an exchange of one unity between states with different parity.This modification is necessary just to reproduce the hierarchy of the masses displayed in Tab.III.Now, the conformal masses for the mesonic hybrids read: For the non-mesonic hybrids they become, Let us refer to this modification as GSWm2.As one can see in Tab.III, the present free parameter approach, is in agreement with the predictions of the recent lattice calculations [39] except for the state 1 −− which is slightly underestimated.Of course, this is just an example of how a simple modification could lead to a good description of the present knowledge of hybrid states.Therefore, let us stress again that the GSW model in general can be also used to provide useful predictions for the Physics of exotic hadrons.For example, we might use the GSW model calculations to try to identify which states, among those addressed in the Particle Data Group, could be considered as hybrid candidates.

VI. PHENOMENOLOGICAL ANALYSIS FOR LIGHT HYBRIDS
Let us discuss our results in light of the experimental data for the spectra [50].We recall that the hybrids states 0 −+ , 0 ++ , 1 −− , 1 ++ have the same quantum numbers as regular mesons, hence one cannot directly consider the latter as hybrid states.However, we will proceed to compare the spectra of heavy mesons, reported in Ref. [50], with our predictions for hybrids assuming that the model will guide us to identify which kind of these states might be hybrids.The corresponding masses are displayed in in Table IV.In the table, we also report the results for both the ground (GSW n=0 ) and the first excited (GSW n=1 ) states obtained from the GSW model.Since Lattice QCD calculations indicate the necessity of an improvement of the present approach, we also include the masses predicted by the simple modification GSWm2 discussed in the previous section.From the comparisons between the theoretical predictions and the experimental data, one might conclude that from the point of view of the model, the following states could be hybrids: 0 −+ : the π(2070) and η(2100) could be consistent with the ground state of a hybrid; 0 ++ : the f 0 (2060) could be the ground state, while the X(2540) could be a first excited state.If future improvements of lattice calculations will confirm the need for the modification of the GSW model, the X(2540) could be also consistent with the ground state of the corresponding hybrid; 1 −− : the φ(2170) and ω(2205) * both have masses consistent with the hybrid ground state; 1 +− : the h 1 (1965) * could be a hybrid ground state; 1 ++ : the a 1 (2095) could be a ground state hybrid.
Let us remark that such a strategy is motivated by the predictive power of this model, as reflected in Refs.[6,16,51].Hence, we might infer that the model provides a reasonable depiction of certain aspects of QCD, and the utilization of its predictions to identify states potentially attributed to hybrids is well justified.TABLE IV: We show the masses of the particles in the PDG whose quantum numbers correspond to mesons [50] and compare them with our calculated values for the hybrids for the first and the second modes.It must be noted that those particles marked with * are presented in PDG outside the summary table.
In Table IV we observe that the masses of the candidate particles, many of which have not been considered or discovered, fall within the range of the lowest mode of our calculation in all cases, and some even fall within the range of the second mode.Upon this analysis, it is clear that finding a pure hybrid state will be challenging, as they are likely to be mixed with mesons.In fact, it is worth to notice that the widths of these states are very large and therefore one might suspect that mixing of states could occur.In future investigations, we will consider applying the same strategy adopted in Ref. [11] where the GSW model has been used to establish the mixing condition between glueballs and meson states.
More interesting are the other quantum numbers which do not correspond to known mesons 0 +− , 0 −− and 1 −+ .No particle appears in the PDG tables for the first two but, for the 1 −+ we have probably one candidate, the π 1 (2015) (see Tab. V).In this case, as one can see, the reported mass is in agreement with the ground state predicted by the GSW model within the experimental error.Let's conclude by noting that when considering the masses in our calculations of the 0 +− and 0 −− states, they should be investigated.However, in this case, special decay properties need to be examined for a distinctive characterization.
We show the masses of the particles in the PDG for the quantum numbers that do not correspond to mesons.The π 1 (2015) has been omitted from the particle table [50].

VII. THE SPECTRUM OF THE NON LIGHT HYBRIDS
Since the holographic approach here adopted relies on conformal symmetry, predictions can be realistic once the chiral symmetry of QCD is restored.Hence, the proposed model does not contain any dependence on the flavor of the constituent quarks of the hadrons.Nevertheless, further modifications of the approach can be taken into account to reproduce the masses of heavy hadrons [12,16].In particular, we apply the approach introduced in section III to s,c and b quark-antiquark pairs.For this purpose, we follow the prescription addressed in Refs.[12,16], namely to add a constant to the mass of the light mesons Let us report the values of the constant C corresponding to the considered quark flavors [12,16]: ,C b = 8700 MeV.We add here also C s associated to strangeness which we has not studied before, C s = 300 MeV.
We then add the above constants to the mass spectra predicted by the GSW model previously calculated, for ground and excited states.We show in Tables VI-VII the results obtained by performing this operation.We also apply the present procedure to the predictions obtained also for ∆ p = 1, i.e., the second modification.In the next sections, we compare the results of this analysis with the predictions of other quark models and lattice QCD calculations.We show the masses obtained for the heavy hybrid interpolating fields defined above, Eqs. ( 9)-( 10) having fixed all our parameters with the scalar hadrons in Ref. [16].We show the masses obtained for the heavy hybrid interpolating fields defined above, Eqs. ( 16)- (17) having fixed all our parameters with the scalar hadrons in Ref. [16].

VIII. LATTICE QCD AND MODEL CALCULATIONS FOR HEAVY HYBRIDS
There have been many calculations interested in the study of heavy hybrid hadrons.In Table VIII we show some of their results [38,[52][53][54] Heavy Hybrid masses LQCD LQCD LQCD SR SR NRQCD NRQCD NRQCD NRQCD (ss) [52] (ss) [52] (cc) [38] (cc) [53] SR( bb) [53] (cc) [54] (cc) [54] ( bb) [54] ( bb [54] 9)-( 10), in refs.[38,[52][53][54] Let us proceed by comparing these spectra with the outcomes of the GSW model (Tab.VI) and its modification (Tabs.VII).Let us start from the lattice evaluation from Refs.[38,52]-The predictions of the model overestimate the ss ground states 0 −+ and 0 ++ .However, our calculations almost agree with the 1 −+ ground and excited states A good agreement is also found for the ground state of the 1 −+ for the cc hadron [38,52].We also compare our results with the model of Ref. [53] (SR).The main differences can be found in the 0 −+ , the 1 −− and the 1 −+ states.Let us conclude with the comparison of the GSW model calculations, with those of Ref. [54](NRQCD).Here one can notice that our predictions o are in line with the outcome of Ref. [54] for the cc and b b states for n = 0.However, the excited states are overestimated.

IX. PHENOMENOLOGICAL ANALYSIS FOR HEAVY HYBRIDS
In this section we will proceed as before.We will use the GSW model to examine the data and determine if they correspond to any states that could potentially be hybrids.In Table IX we present particle masses from the PDG [50] with mesonic quantum numbers also shared with possible heavy hybrids, and we compare them with the first two modes of our calculation.
In the ss case, the prediction of the GSW model overestimates, for example, the φ(2170) state.For the cc hadrons, the χ c0 (4500), the Ψ(4660) and the Z c (4430) states are possible candidates.Finally, for b b hadrons the χ b0 (2P ), the Υ(10860), the Υ(11020) are close to the spectra predicted by the GSW model.The Z b (10650) and the χ b1 (3P ) are overestimated but, the disagreement is not particularly big.

X. CONCLUSIONS
We have used the Graviton Soft Wall model, previously developed for conventional hadrons, to analyze hybrid hadrons.To this aim, we have computed the conformal masses by examining the potential minimal p-form field configuration that can be obtained from quark and gluon fields.These configurations define the spins and parities of the hybrids and determine the corresponding conformal masses, which in turn characterize the corresponding bound state equations in the fifth dimension.It is important to highlight that our calculation is parameter free, as the two parameters employed in the approach have been determined by the scalar glueballs and mesons [12,16].Additionally, for the heavy hybrids, we have included the same parameters as those used for heavy mesons, which essentially represent the heavy quark masses.The results obtained in our parameter free calculation has been compared with other model calculations.We have analyzed in detail the similarities and differences.We tend to agree with the lattice results and the NRQCD better than with the SR approach.Moreover, we found out that the GSW model, due to the simplicity lowest order p-forms, leads to the same conformal mass for different states and therefore to mass degeneracies.Looking back at table I we see degeneracies between the 0 −+ , the 0 ++ , the 0 +− and the 0 −− hybrids, and the 1 −− , the 1 +− , the 1 ++ and the 1 −+ hybrids.In order to remove such a degeneracy, not predicted by recent lattice calculations, we took into account the effeccts of anomalous dimensions for some of these state and,

TABLE IX:
We show the masses of the particles in the PDG whose quantum numbers addressed in the present analysis but corresponding to mesons [50] and compare them with the results of the calculattions of the heavy hybrid spectra with the GSW model and its modification (GSW m2 ).
hence, the corresponding degeneracies are eliminated, as can be seen in Table III for the two modifications studied.The remaining degeneracies correspond to underlying symmetries not accounted for in QCD.Therefore, additional adjustments to the naive GSW model should be pursued for more accurate predictions.Finally, we have proposed a different approach to analyze the PDG spectra.We assume our model's results are accurate mass values for hybrids and attempt to identify potential states in the data using these mass values, indicating the possible presence of hybrids in the spectra.In this way we have predicted several possible hybrid states but, in many cases given the closeness to conventional mesons states, one expects strong mixing between mesons and hybrids.

TABLE I :
The hybrid masses obtained from Eqs. (

TABLE II :
We show the masses obtained for the hybrid hadrons, whose quantum numbers are defined above in Eqs. (

TABLE III :
We show the masses obtained for the hybrid hadrons, whose quantum numbers are defined above in Eqs. (