Meson and Glueball spectroscopy within the Graviton Soft-Wall model

In this contribution we present results of the calculations of several hadronic spectra within the holographic graviton soft-wall (GSW) model. In particular, we studied and compared with data for the ground state and excitations of: glueballs, scalar, vector, axial and pseudo-scalar mesons. The GSW model is found to be capable to describe these observable with only few parameters.


Introduction
In this contribution we investigate the spectra of some hadronic species. We consider the graviton soft-wall (GSW) model, introduced and applied in Refs. [1][2][3][4], initially adopted to describe non perturbative features of glueballs. We remind that the holographic approach relies in a correspondence between a five dimensional classical theory with an AdS metric and a supersymmetric conformal quantum field theory. Since the latter is not QCD, we use the socalled "bottom-up" approach [5][6][7]. where the five dimensional classical theory is properly modified to reproduce nonperturbative QCD properties as much as possible. Furthermore, the GSW model is a modification of the initial softwall (SW) where a dilaton field is introduced to softly break conformal invariance. In GSW model, in order to properly describe the scalar glueball spectrum, a modification of the metric has been proposed. This model has been successfully applied to reproduce non perturbative features of mesons and glueballs [1,[7][8][9][10]. In particular, we calculated and impressively described the spectra of: glueballs (with even and odd spin), the light and heavy scalar mesons, the ρ, the a 1 , the η and the π. Moreover, we showed in Ref. [3] that only when the masses of the glueballs and the mesons are close, mixing is to be expected [11]. However, if this mass condition is associated to a different dynamics, mixing will not happen [12].

Essential features of the GSW model
The essential difference between the GSW model from the traditional SW one, is a deformation of the AdS metric in 5 dimensions: where g M N is the AdS 5 metric and φ 0 (z) = k 2 z 2 [1, 8-10, 13, 14]. Modifications of the metric have been also pro-posed in other studies of the properties of mesons and glueballs within AdS/QCD [8,[14][15][16][17][18][19][20][21][22]. The action, in the gravity sector, written in terms of the standard AdS metric of the SW model, is: In the GSW model, the parameter α encodes the effects due to the modification of the metric, while, β is used to recover the SW results as much as possible. Indeed, β is not a free parameter and it is fixed to lead the SW kinematic term in the action [1,3,4]. For example, for scalar fields β = β s = 1 + 3 2 α and for a vector β = β v = 1 + 1 2 α. In Ref. [2], an additional dilaton φ n has been also phenomenologically proposed to describe scalar and pseudo-scalar mesons in order to obtaina binding potential in the equation of motion. We anticipate that this quantity does not contain any free parameter, which are namely α and k. In closing, in order to properly take into account the chiral symmetry breaking, essential to describe the pion spectrum, a modification of the dilaton has been proposed.

The glueball spectra within the GSW model
In this section we discuss and present the GSW predictions for the glueball spectra with even and odd spins together with the successful comparison with lattice data.

Scalar glueballs
Here we recall the main results discussed in Ref. [1]. We remind that in this case, the GSW model predicts that the scalar glueball is described by its dual graviton which is a solution of the Einstein equation (Ee) for a perturbation the metric (1 (3) where, as usual, we assumed factorization between the z and x µ dependence, t = αk 2 /2 z and Λ 2 = (2/αk 2 ) M 2 , being M the mode mass.
It is remarkable that the potential is uniquely determined by the metric and its modification. The only free parameter is the scale factor depending on αk 2 . This term is fixed from the comparison with lattice QCD [1]. As one can see in the left panel of Fig. 1, for αk 2 ∼ (0.37 GeV) 2 the linear glueball spectrum is well reproduced, at variance with the SW model. We also stress the good agreement with the graund state mass obtained by the BESIII data of the J/Ψ decays [23,24] (SDTK) very recently, after analysis.

Spin dependent glueball spectra
We found out that the ground state of glubealls with spin is well reproduced if we consider the approach of Refs. [8,20,25] to describe spin effects. In this case the action is that of a scalar field [4]: and the spin dependence is encoded in the 5-dimensional mass term: i) M 2 5 R 2 = J(J + 4) for even spin J and ii) M 2 5 R 2 = (J + 2)(J + 6) for odd spin J. One should notice that since M 2 5 R 2 ≥ 0, the potential in equation of motion is binding and therefore no additional dilatons are needed. Results of the calculations for the odd and even glueballs are shown in Tables I-II, respectively. As one can see, results are in fairly agreement with data. We also evaluated and compared the Regge trajectories provided by the GSW model with lattice data. In general, the form is J ∼ a g M 2 + b g , where g = o stands for odd spin and g = e is referred to even spin, respectively. In the odd case: a o = 0.18 ± 0.01, b o = −0.75±0.28 [2], in agreement with J ∼ 0.18M 2 +0.25 [26]. For even glueballs: a e = 0.21 ± 0.01, b e = 0.58 ± 0.34 [2] in agreement with J ∼ 0.25M 2 [27,28].

Meson spectroscopy
Here we show the main results for the spectroscopy of the f 0 , heavy scalar, ρ, a 1 , η and pion mesons. We stress again that for the latter case a modification of the dilaton φ 0 must be included to incorporate chiral symmetry breaking.

Light and heavy scalar mesons
In this case the action is that of Eq. (4) but since now M 2 5 R 2 = −3, the relative potential is not binding. Therefore, at variance to the the glubeall case, the additional contribution exp[−φ n (z)] must be included. Details on this topic are presented in Ref. [2]. Here we mention that φ n is chosen to produce in the potential a term proportional to the expansion, up to the second order, of exp(αk 2 z 2 ) and thus preserving the binding feature. By keeping fixed αk 2 = 0.37 GeV 2 , we found a reasonable good fit, see left panel of Fig. 1, for 0.51 ≤ α ≤ 0.59. In the case of heavy mesons we added the quark mass contribution to the light scalar masses [2,4] in order to effectively include the dynamics the mass of the heavy quarks [29][30][31]. In particular, the heavy mass (M h ) is obtained from the previous light scalar mass (M l ) as follows: M h = M h + C, where C is the contribution of the quark masses. C c = 2400 MeV, for the cc mesons, and for the bb mesons C b = 8700 MeV. The successful comparison with data [4] is displayed in the left and right panels of Fig.  1. One should notice that C c and C b are comparable with the values of 2m c and 2m b , respectively, as expected.

The ρ spectrum
For the ρ meson, the action is equal to that of a vector field within the usual SW model since M 2 5 R 2 = 0 [20]: (5) this case, there is no need of the auxiliary dilaton since M 2 5 R 2 = 0 and thus the potential is binding. As one can see in the left panel of Fig. 2, the agreement is good, exception is ρ(770). Such a discrepancy suggests that the GSW model must be further improved.

The a 1 axial meson spectrum
In the case of the axial-vector mesons, due to chiral symmetry breaking, M 2 5 R 2 = −1 [32,33]. Therefore the EoM for the a 1 can be obtained from the action of a vector field with a conformal mass different from zero: Since in this case M 2 5 R 2 < 0, the corresponding potential is not binding and a modification of the dilaton is required. Details on the differential equation defining the contribution are included in Ref. [2]. With the parameters previously addressed, we get the spectrum shown in the central panel of Fig.2. Our calculation favors that the a 1 (1930), a 1 (2095) and a 1 (2270) are axial resonances [34]. Moreover, the agreement is even more impressive if a missing ground state with a mass lower then the quoted 1230 MeV will be observed.

The η pseudo-scalar meson
The EoM is similar to that of the scalar case but now, M 2 5 R 2 = −4 [33]. As expected, one needs to include the additional dilaton to get a binding potential. In the right panel of Fig. 2 we show our calculation of the spectrum, we remind that the band stands for the theoretical error on α α = 0.55 ± 0.04. The comparison with the experimental data [35,36] is very good also in this case. Moreover, the GSW model predicts that η(1405) and η(1475) are degenerate, as discussed in PDG review. Moreover, since in the upper mass sector the experimental mass gap is larger, the GSW model also favors: i) the existence of two resonances between the η(1760) and the η(2225) and ii) that the η(1405) and η(1470) are the same resonance. We recall that the results for the ρ, a 1 and η masses are free parameter calculations.

The pion spectrum
In the case of the pion, the Goldstone boson of SU(2) x SU(2) chiral symmetry, as already anticipate, we need to incorporate in the model chiral symmetry breaking mechanism. Since, as discussed in e.g., Refs. [9,15,37], the physics of confinement and chiral symmetry breaking could ascribed to the dilaton [9,15,37], we propose a modification of the dilaton profile function [2]: we consider: The parameter tanh (δ) is responsible for the the chiral symmetry breaking, see details in Ref. [2]. This choice preserves the large z behaviour, which leads to Regge trajectory of the higher mode spectrum. On the other hand, the low z region describes the transition region and δ and γ incorporates the effects of the spontaneous chiral symmetry breaking. Also in this case, in analogy with the η, we need to include φ n to get a binding potential. The pion spectrum is shown in the Table III compared with the PDG data [35,36]. The predicts more pion states the experimentally observed [33,37].

Glueball-Meson mixing
Here we discuss the conditions for a not favorable mixing, i.e. states with mostly gluonic valence structure [3]. We consider an holographic light-fron (LF) representation of the EoM in term of the Hamiltonian [5] and a two dimensional Hilbert space generated by a meson and a glueball states, {|Ψ m >, |Φ g >}. Mixing occurs when the hamiltonian is not diagonal in the subspace. A matrix representation of the hamiltonian is given by where α =< Ψ m |H|Φ g >, m 1 =< Ψ m |H|Ψ m > and m 2 =< Φ g |H|Φ g >. We are assuming m 2 > m 1 and for simplicity α real and positive. After diagonalization the eigenstates have a mass M ± = m ± α 2 + (∆m) 2 , where m = (m 1 + m 2 )/2 and ∆m = (m 2 − m 1 )/2. The first physical meson, assuming to be the lightest one, is given by the eigenvector of H [3]. Since we fixed the meson spectrum to the experimental values, |Ψ phy > represents a physical meson state while we have fixed the glueball spectrum to the lattice values, therefore the glueball state is our initial state |Φ g >, thus The mixing probability is proportional to the overlap of these two wave functions (w.f.). We calculate the probability for no mixing, i.e., P GM = 1 − | Ψ phy |Φ g | 2 . As one can see in Fig. 3, the the mixing should occur when n g = 2, 3, 4 and the meson mode numbers n ∼ 10, 13, 17. This condition reduces the overlap probability for mixing dramatically. Therefore, we predict the existence of almost pure glueball states, in the scalar sector, in the mass range above 2 GeV.

Conclusions
In this contribution we presented the applications of the GSW model to the glueball and meson spectra. We saw that the proposed modification of the metric is fundamental to reproduce experimental data with only two parameters. We propose the inclusion of an additional free parameter dilaton to get binding potential for tachionic 5-dimensional masses. Excellent agreements with data are found. For the pion, the SW dilaton has been properly modify to describe the chiral symmetry breaking in the model. Also in this case the comparison with data is quite good. We conclude by remarking the capability of the model in reproducing several masses of very different hadronic systems with only few universal parameters and therefore leading to a relevant predicting power.