Deploying heavier η meson states: configurational entropy hybridizing AdS/QCD
DDeploying heavier η meson states: configurational entropy hybridizing AdS/QCD R. da Rocha ID ∗ Federal University of ABC, Center of Mathematics, Santo Andr´e, 09580-210, Brazil
The meson family of η pseudoscalars is studied in the context of the AdS/QCD correspondenceand the differential configurational entropy (DCE). For it, two forms of configurational-entropicRegge-like trajectories are engendered, relating the η mesonic states excitation number to boththeir experimental mass spectrum in the Particle Data Group (PDG) and the DCE as well. Hence,the mass spectrum of η pseudoscalar mesonic states, beyond the already detected states η (550), η (1295), η (1405), η (1475), η (1760), η (2225), and η (2320), is derived for any excitation number.The three first ulterior members of this family are then analyzed and also compared to existingcandidates in PDG. PACS numbers: 89.70.Cf, 11.25.Tq, 14.40.Aq
I. INTRODUCTION
Shannon’s information entropy paradigm resides inconverting information into a coded form in random sys-tems. Compressed messages do present the same amountof information as the original ones. Nevertheless, theyare less redundant [1]. In his seminal work, Shannon re-placed information, from an unestablished concept, intoa precise theory that has underlain the digital revolu-tion. It has inspired the development of all moderndata-compression algorithms and error-correcting codesas well. Information entropy is the core of formulatingthe configurational entropy (CE) and consists of the partof the entropy of a system that is related to representa-tive correlations among its finite number of integrants.The CE of any system evaluates the rate at which infor-mation is condensed into wave modes. For the continu-ous limit, the differential configurational entropy (DCE)takes place and can be applied to a considerable varietyof problems when one introduces some distribution por-traying the system that is related to physical quantitiesunder scrutiny [2–4].The DCE merges informational and dynamical con-stituents of physical systems [5]. The DCE logarithmi-cally evaluates the number of bits needed to designatethe organization of the studied system. Employing theDCE to approach a system requires a scalar field that isspatially localized. One typically uses the energy density,as the T component of the energy-momentum tensor.Other choices, like scattering amplitudes, are also suit-able, according to the system to be approached [6, 7].The DCE was also employed to distinguish dynamicalsystems among regular, entirely random, and also chaoticevolution in QCD, providing observables and techniquesto study physical phenomena [8]. The DCE setup addi-tionally refined the comprehension of phenomenologicalAdS/QCD, playing the color glass condensate an impor-tant role on studying meson states [9–11]. With the DCE ∗ Electronic address: [email protected] tools, heavy-ion interactions were else investigated [8].The DCE has been systematically shown to be a veryimportant criterion to examine some of the most essen-tial features of AdS/QCD.The DCE estimated new features of light-flavormesonic states of several families in AdS/QCD [4, 12, 13],corroborated by several experiments reported in Parti-cle Data Group (PDG) [14]. Besides, charmonia andbottomonia, tensor mesons, scalar glueballs, odderons,pomerons, and baryons, amongst other particle states,were scrutinized in the DCE and AdS/QCD contexts,also incorporating finite temperature [15–20]. A varietyof leading applications of the DCE have been also inves-tigated in Refs. [21–27]. Besides, the DCE is decisivefor investigating the gravity point of view in AdS/QCDduality. Black branes were studied in Ref. [28] in the con-text of the DCE, whereas black holes and thermal AdSbackgrounds yielded a temperature-dependent DCE inRef. [29]. Also, Bose-Einstein condensates in AdS/CFT,modeling self-interacting gravitons, were explored underthe DCE apparatus [30].The AdS/QCD framework is constituted by an AdSbulk wherein gravity, which is weakly coupled, emulatesthe dual setup to the four-dimensional (conformal) fieldtheory (CFT). In the duality dictionary, physical fieldsin the AdS bulk are dual objects to boundary operatorsof (strongly coupled) QCD. The bulk fifth dimension isnothing more than the energy scale of QCD. Confine-ment can be, hence, achieved by either by a Heavisidecut-off in the bulk, the hard wall, or by a dilatonic field,that accomplishes a smooth cut-off along the AdS bulk –the soft wall model [31, 32]. The soft wall model assertsthat Regge trajectories naturally emerge from confine-ment when one employs a quadratic dilaton in the actionthat governs the bulk fields. Although the soft wall modelsucceeds in some aspects, it does not address other rele-vant aspects of QCD. Those aspects that are not encom-passed by the soft wall include the derivation of mesonicdecay constants, the chiral symmetry breaking (CSB)and the quarkonium mass spectrum, among others [33].A decisive answer to these issues yielded a ultraviolet(UV) cut-off to be considered [34], permitting an energyscale z sitting along the bulk. Although pseudoscalar a r X i v : . [ h e p - t h ] F e b mesons were scrutinized within a non-perturbative setup,in AdS/CFT [35], here a modified AdS/QCD model willbe employed, where η pseudoscalar mesons are dual ob-jects to free bulk fields, having an appropriate anomalousdimension that gauge the bulk field mass [36].In this work, the DCE for the η pseudoscalar mesonsfamily will be computed, as a function of both the η me-son family excitation number and mass spectrum. Theinterpolation curves to these data engender two formsof DCE Regge-like trajectories. Hence, the mass spec-trum of η meson states, with higher excitation numbers,can be then obtained, consisting of the next generationof η mesons to be experimentally detected. This arti-cle is displayed as follows: Sec. II devotes to reviewthe AdS/QCD duality. From the CSB and the intro-duction of the anomalous dimension, the mass spectrumof η pseudoscalar mesons family is derived, with the aidof bulk-to-boundary propagators and correlators. In Sec.III, the fundamentals of the DCE are introduced and dis-cussed, being the DCE computed for the η pseudoscalarmeson family. The DCE is then expressed as a function ofthe η mesons n excitation number and their experimen-tal mass spectrum as well. Therefore, two forms of DCERegge-like trajectories of η mesons consist of the resultinginterpolation curves for these two functions, predictingthe mass spectrum of heavier η mesonic states, which docorrespond to higher n mesonic η states. Sec. IV encom-passes an overall analysis of the results and conclusions. II. ADS/QCD ESSENTIAL FRAMEWORK
The starting point is the model introduced in [34]. Onedefines the AdS Poincar´e patch with UV cut-off as ds = g AB dx A dx B (1)= R z (cid:0) dz + η µν dx µ dx ν (cid:1) Θ ( z − z ) , for Θ ( x ) being the Heaviside step function, R is the AdSradius, and z is the UV cut-off defined by the geomet-rical locus of a D-brane. Choosing the AdS boundaryyields the conformal invariance to be naturally brokenby the energy scale z [34]. This model has been consis-tently employed in studying, in particular, meson fami-lies in AdS/QCD [37]. Making the conformal boundary z → Φ ( z ) = k z .Pseudoscalar mesons can be described by a function P ( x µ , z ), and the respective action reads S P = − g P (cid:90) d x √− gL, (2)where the Lagrangian is given by L = e − Φ ( z ) (cid:2) g AB ∂ A P ∂ B P + M P (cid:3) , (3)The constant g P fixes the units of the action with re-spect to the number of colors, N c , of the CFT, whereas M stands for the bulk mass of pseudoscalar mesons.The action (3) yields the bulk fields equations of motion.According to the gauge/gravity setup, scalar bulk fieldssolutions scale as z ∆ − in the UV regime, being dual toan O operator in the boundary CFT. Its associated 2-point correlator reads [38] (cid:104)O ( x ) O ( ) (cid:105) ∝ | x | − ∆ , (4)where ∆ denotes the scaling dimension of O . The latter ∆ term is also related to the operators that create hadrons.Mesons can be thus implemented when one employs q ¯ q operators, corresponding to ∆ = 3. As bulk mass termsin the action affect the UV limit of the solutions, there-fore ∆ , the term M , and the spin s are coupled to as[38] M R = ∆ ( ∆ − − s ( s − . (5)Scalar mesons correspond to M R = −
3, whereas vec-tor mesons satisfy M R = 0. Eq. (5) holds for the s -wave, (cid:96) = 0, approach of mesons. For (cid:96) (cid:54) = 0 states, atwist-like operator, τ , must modify the conformal dimen-sion as ∆ (cid:55)→ ∆ + τ [39–41].The introduction of an anomalous dimension yields theexploration of other features of mesons. Indeed, mesonshappen to be non-degenerate states, after CSB. For dis-tinguishing them, the anomalous dimension, ∆ P , identi-fies the parity of the mesonic state, here to be consideredthe η pseudoscalar family. It promotes, hence, a CSBmechanism, by adjusting the bulk mass as M = 1 R ( ∆ + ∆ P ) ( ∆ + ∆ P − − s ( s − , (6)where ∆ emulates an effective anomalous dimension, asproposed in details in Ref. [33]. Employing Eqs. (2, 3),one can derive the mass spectrum of the η mesonic states.Under parity, J P C = 0 − + , the η mesons Regge trajec-tory is not invariant. Ref. [33] used the Breitenholner–Freeman limit to show that ∆ P = −
1, for the η mesonfamily. Besides, the inequality M R ≥ − η pseudoscalarmeson family is obtained by varying the action (2), yield-ing z ∂∂z (cid:32) e − k z z ∂P∂z (cid:33) + e − k z (cid:0) P − z (cid:3) P (cid:1) = 0 . (7)One implements the Fourier transform with respect tothe q µ -momentum space, P ( q, z ) = 1(2 π ) (cid:90) (cid:90) (cid:90) (cid:90) Boundary d q e ix µ q µ P ( x µ , z ) , (8)denoting P ( q, z ) = ˜ P ( q ) p ( q, z ) , where ˜ P ( q ) stands for thesources and p ( q, z ) is the pseudoscalar bulk-to-boundarypropagator, with condition P ( q, z ) = 1. Therefore, Eq.(7) yields z ∂ p∂z − (3 + 2 k z ) p ∂p∂z + (cid:0) z q + 4 (cid:1) p = 0 . (9)Following well established holographic procedures formesons [31, 41, 42], the mass spectrum can be derived.Eq. (9) is a Kummer differential equation, whose solu-tions are expressed by the Kummer confluent hypergeo-metric function, F ( a, b, x ), as p ( q, z ) = z F (cid:0) − q , , k z (cid:1) z F (1 − q , , k z ) , (10)where q ≡ q k . The on-shell boundary action in the pseudoscalar caseis obtained by evaluating the solution (10) into Eq. (3): I η Boundary = R g S (cid:90) (cid:90) (cid:90) (cid:90) Boundary d q (2 π ) (cid:34) e − k z z ˜ P ( q ) ˜ P ( − q ) p ( q, z ) ∂p ( z, − q ) ∂z (cid:12)(cid:12)(cid:12) z = z (cid:21) . (11)Therefore the correlator, Π η ( q ) = δ I η Boundary δ ˜ P ( − q ) δ ˜ P ( q ) , reads [34]Π η (cid:0) q (cid:1) = R e − k z g S z ∂p ( q, z ) ∂z (cid:12)(cid:12)(cid:12) z = z . (12)Replacing Eq. (10) into (12) yieldsΠ η (cid:0) q (cid:1) = − (cid:34) z +(1 − q ) 2 k z F (cid:0) − q , , k z (cid:1) F ( q , , k z ) (cid:35) × R e − k z g S z . (13)As one regards the on-shell mass condition, q = M n ,the η mesons family mass spectrum can be read of thepoles of (13), corresponding to the roots ξ n ≡ M n / k of the Kummer function in the denominator of Eq. (13),namely [43], F (cid:0) − ξ n , , k z (cid:1) = 0 . Hence the η mesonsmass spectrum reads M n = 4 k ξ n ( k, z , ∆ P ) . (14)The numerical results are presented in the fourth columnof Table I, when taking k = 0 .
450 GeV and z = 5 . − . η meson states n × × × × × m MeV Experimental η meson mass spectra FIG. 1: Experimental squared mass spectrum of the η pseu-doscalar meson family, for n = 1 , . . . ,
9, with error bars, withstates η (550), η (cid:48) (958), η (1295), η (1405) ∼ η (1475), η (1760), η (2225), η (2320) in PDG [14].———– η pseudoscalar mesons mass spectrum ———— n State M Experimental (MeV) M Theory (MeV)1 η (550) 547 . ± .
017 975.22 η (cid:48) (958) 957 . ± .
06 1011.13 η (1295) 1294 ± η (1405) /η (1475) 1408 . ± . η (1760) 1751 ±
15 1829.16 η ( (cid:3) ) 1992.67 η ( (cid:3) ) 2087.38 η (2225) 2221 +13 − η (2320) 2320 ±
15 2289.4TABLE I: Both the experimental and the AdS/QCD-predicted mass spectrum of the η pseudoscalar meson family.The identification of the η particle states to their respective n excitation numbers follow the seminal work [44]. As discussed in Sec. 63 of PDG 2020 [14], the η (1405) andthe η (1475) states might be the same particle [45, 46] andhere this identification is implemented. The η ( (cid:3) ) statewith n = 6, in Table I, might be identified to the η (2010),with experimental mass 2010 +35 − MeV, whereas the η ( (cid:3) )state with n = 7 might be identified to the η (2100), withexperimental mass 2100 +30+75 − − MeV [14]. The η (2100) isargued to be alternatively interpreted as a pseudoscalarglueball or just a Regge excitation of η (cid:48) . For more detailssee, e. g., Refs. [47–51]. III. DCE REGGE-LIKE TRAJECTORIES OF η MESONS
The DCE evaluates correlations amongst the fluctua-tions of the energy configurations into the physical sys-tem to be studied. To portray the system, the en-ergy density – the time component of the stress-energy-momentum tensor T ( r ) – is the essential ingredient,where r ∈ R m , for any finite dimension m . The correla-tor Π( r ) = (cid:90) R m d m x T (˚ r + r ) T (˚ r ) (15)defines the probability distribution that determines theDCE as the Shannon’s (information) entropy of correla-tions [16]. For constructing the DCE, one first calculatesthe Fourier transform T ( k ) = 1(2 π ) m/ (cid:90) R m d m x T ( r ) e − i k · r . (16)A detector identifies a wave mode, within a volume d m k centered at k , with a probability that is propor-tional to the power spectrum in that mode, p ( k | d m k ) ∼| T ( k ) | d m k [2]. Eq. (16) can be also seen as a proba-bility distribution in momentum space for different wave-lengths that impart the generation of correlations acrossthe system. This probability distribution engenders themodal fraction [5], T ( k ) = | T ( k ) | (cid:82) R m d m k | T ( k ) | . (17)The amount of information to describe T , with respectto wave modes, is computed by the DCE,DCE T = − (cid:90) R m T (cid:63) ( k ) ln T (cid:63) ( k ) d m k , (18)where T (cid:63) ( k ) = T ( k ) / T max ( k ), and T max ( k ) is the supre-mum of T in R m wherein the power spectrum peaks.The DCE has units of nat/unit volume . The DCE en-codes a scale information, as the power spectrum is rep-resented the Fourier transform of the 2-point correlator[2].As we are here scrutinizing the η pseudoscalar mesonfamily, the value m = 1 is taken into account in Eqs.(16 – 18), representing the scale energy, z , along the AdSbulk. Replacing the Lagrangian (3) into the time com-ponent of the energy-momentum tensor, T = 2 √− g ∂ ( √− gL ) ∂g − ∂∂x γ ∂ ( √− gL ) ∂ (cid:16) ∂g ∂x γ (cid:17) , (19)the expectation value of the energy density, to be usedin Eqs. (16 – 18), reads (cid:104) T ( z ) (cid:105) = 1 z (cid:2) P (cid:48) ( z ) + M P ( z ) (cid:3) . (20) Nat denotes the natural unit of information. One nat equals1 / ln 2 bits, representing the information that underlies a uniformdistribution defined on the real range [0 , e ]. With the energy density (20) of the η pseudoscalar mesonfamily in hands, an alternative procedure for derivingtheir mass spectrum is put into action, solely using theDCE. It is worth to emphasize that this method, thathybridizes AdS/QCD, can be realized as a more accurateone, when compared to pure AdS/QCD predictions. Infact, it is based on the interpolation of the η meson familyexperimental mass spectrum in PDG [14].The DCE, using the protocol (16 – 18), for the η pseu-doscalar meson family, can be numerically computed andthe results are shown in Table II. n State CE1 η (550) 3.9942 η (cid:48) (958) 6.2123 η (1295) 8.2484 η (1405) ∼ η (1475) 9.9435 η (1760) 14.5406 η ( (cid:3) ) 23.8347 η ( (cid:3) ) 28.8218 η (2225) 35.5619 η (2320) 42.03510* η η η η pseudoscalar meson family.States with an asterisk represent the members of the η me-son family whose mass is obtained by the DCE Regge-liketrajectories (21, 22), respectively for the excitation numbers n = 10 , , Standard Regge trajectories AdS/QCD show the light-flavor meson spin to be proportional to the square of themass spectrum. One can then emulate them, using theintrinsic DCE of the η pseudoscalar meson family. Forit, the DCE of the η pseudoscalar mesonic states canbe regarded as a function of the η mesons mass spectrumthat have been already detected in experiments [14]. Theinterpolation curve of this data consists of the secondform of DCE Regge-like trajectories, whereas the firstform takes into account the DCE as a function of the n excitation modes of η meson states. Fig. 2 showsthe corresponding results, wherein numerical interpola-tion yields the first form of DCE Regge-like trajectory.The explicit expression is numerically obtained by inter-polation of data in Table II:DCE η ( n ) = − . n +0 . n − . n +4 . n + 1 . , (21)within ∼ .
5% standard deviation. η mesons n η ( n ) FIG. 2: DCE of the η pseudoscalar meson family, for n = 1 , . . . , η (550), η (cid:48) (958), η (1295), η (1405) ∼ η (1475), η (1760), η (2225), η (2320) in PDG [14]and to the two states η ( (cid:3) )) as a function of n . The first formof DCE Regge-like trajectory is plotted as the gray dashedline. η mesons × × × × × m ( MeV ) η ( m ) FIG. 3: DCE of the η pseudoscalar meson family, for n = 1 , . . . , η (550), η (cid:48) (958), η (1295), η (1405) ∼ η (1475), η (1760), η (2225), η (2320) in PDG [14]and to the two states η ( (cid:3) )) as a function of their (squared)mass. The second form of DCE Regge-like trajectory is plot-ted as the gray dashed line. As there is no known experimental data associatedwith the η ( (cid:3) ) states, the hybrid interpolation in Fig.3 takes the experimental masses of η (550), η (cid:48) (958), η (1295), η (1405) ∼ η (1475), η (1760), η (2225), η (2320)in PDG [14] and the AdS/QCD-predicted masses of thetwo states η ( (cid:3) ), as in Table I. As already pointed out,one might try to identify the η ( (cid:3) ) state with n = 6 to η (2010), with mass 2010 +35 − MeV, and the η ( (cid:3) ), n = 7,state to η (2100), whose mass equals 2100 +30+75 − − MeV[14]. However, as the η (2100) may be a pseudoscalarglueball or a Regge excitation of η (cid:48) , taking the η (2100)mass in PDG for deriving the DCE Regge-like trajecto-ries is too speculative. Anyway, either using the hybridinterpolation or the interpolation with the experimentalmasses of η (2010) and η (2100) yields a tiny differenceof 0 .
1% in the derived masses of η , η , η . Hence, as both methods provide similar masses of the η , η , η states up to 0 . η mesons mass, m (MeV), is listed as follows:DCE η ( m ) = 1 . × − m + 1 . × − m +7 . × − m − . × − m +5 . , (22)within ∼ .
17% standard deviation. The DCE Regge-liketrajectories in Eqs. (21, 22) illustrate several importantfeatures about the η pseudoscalar meson family. Using(21, 22) makes one to compute the DCE of membersof the η pseudoscalar meson family, for n ≥
10. Theyconsist of the first next states of the η pseudoscalar mesonfamily to be detected in experiments.As illustrated in Table II, replacing n = 10 in Eq.(21) yields the DCE equal to 48.474. Then, substitutingthis value in the left-hand side of Eq. (22), and solvingfor m , one obtains for the tenth member, η , of the η pseudoscalar meson family, the mass m η = 2428 . n = 11, one derivesthe mass of the eleventh member, η , as m η = 2533 . X (2632) meson, having mass m = 2635 . ± . η . Besides, replacing n = 12 in Eq. (21) yieldsthe DCE equals to 58.464. Therefore, putting back thisvalue into the left-hand side of Eq. (22), solving for m yields the mass m η = 2695 . η memberof the η pseudoscalar meson family. One can speculatethat the η mesonic state corresponds to the alreadydetected X (2680), with mass 2676 ±
27 MeV [14].
IV. CONCLUDING REMARKS
We regarded a useful technique, hybridizing DCE toAdS/QCD, to derive the mass spectrum of η pseudoscalarmeson states. with higher excitation number. For it, thequadratic dilaton model with anomalous dimension wasemployed. The compactness of DCE methods, used inSec. III, is here advantageous for at least two main rea-sons. The first one is the avoidance of computationalissues that are common in approaching AdS/QCD phe-nomenology. The second one resides in hybridizing DCEto AdS/QCD. In this case, obtaining the mass spectrumof the next generation of η mesons to be detected takesinto account the nine first η mesons masses that were ex-perimentally detected [14], as illustrated in Fig. 3 andanalytically approximated by Eq. (22) with a very goodaccuracy. With the aid of data in Fig. 2 and Table II, an-other form of DCE Regge-like trajectory (21) completesthe necessary ingredients for deriving the mass spectrumof η pseudoscalar meson states. The derived mass spec-trum agrees with meson states candidates, recently listedin PDG 2020 [14], as already comprehensively discussedin the last paragraph of Sec. III.According to Table II and Fig. 2, η pseudoscalar me-son states with lower n excitation number have higherconfigurational stability, from the DCE point of view.Lower values of the DCE can point to more predomi-nant η meson states. Similarly, the more massive the η pseudoscalar meson state, the higher its configurationalinstability is. Hence, more massive η pseudoscalar me-son states are more unstable from the point of view ofthe DCE, being also less predominant and prevalent η meson states. This interpretation also corroborates withthe number of events, that characterizes each η meson, already detected by experiments [14]. Besides the pecu-liarities of producing each η meson as a by-product ofprecise interactions and reactions, the DCE might alsosuggest the phenomenological prevalence of states in the η pseudoscalar meson family. Acknowledgments:
RdR is grateful to FAPESP(Grant No. 2017/18897-8) and to the National Coun-cil for Scientific and Technological Development – CNPq(Grants No. 303390/2019-0 and No. 406134/2018-9), forpartial financial support. [1] C. E. Shannon, Bell Syst. Tech. J. (1948) 379.[2] M. Gleiser, M. Stephens and D. Sowinski, Phys. Rev. D (2018) 096007 [arXiv:1803.08550 [hep-th]].[3] M. Gleiser and N. Stamatopoulos, Phys. Lett. B (2012) 304 [arXiv:1111.5597 [hep-th]].[4] A. E. Bernardini and R. da Rocha, Phys. Lett. B (2016) 107 [arXiv:1605.00294 [hep-th]].[5] M. Gleiser and N. Stamatopoulos, Phys. Rev. D (2012) 045004 [arXiv:1205.3061 [hep-th]].[6] G. Karapetyan, Phys. Lett. B (2018) 201[arXiv:1802.09105 [nucl-th]].[7] G. Karapetyan, Phys. Lett. B (2018) 418[arXiv:1807.04540 [nucl-th]].[8] C. W. Ma and Y. G. Ma, Prog. Part. Nucl. Phys. (2018) 120 [arXiv:1801.02192 [nucl-th]].[9] G. Karapetyan, EPL (2017) 18001 [arXiv:1612.09564[hep-ph]].[10] G. Karapetyan, EPL (2017) 38001 [arXiv:1705.10617[hep-ph]].[arXiv:1912.10071 [hep-ph]].[11] G. Karapetyan, EPL (2019) 58001 [arXiv:1901.05349[hep-ph]].[12] A. E. Bernardini and R. da Rocha, Phys. Rev. D (2018) 126011 [arXiv:1809.10055 [hep-th]].[13] L. F. Ferreira and R. da Rocha, Phys. Rev. D (2019)no.8, 086001 [arXiv:1902.04534 [hep-th]].[14] P. A. Zyla et al. (Particle Data Group), Prog. Theor.Exp. Phys. (2020) 083C01.[15] L. F. Ferreira and R. da Rocha, Phys. Rev. D (2020)106002 [arXiv:1907.11809 [hep-th]].[16] N. R. F. Braga, L. F. Ferreira and R. da Rocha, Phys.Lett. B (2018) 16 [arXiv:1808.10499 [hep-ph]].[17] N. R. Braga and R. da Mata, Phys. Rev. D (2020)105016 [arXiv:2002.09413 [hep-th]].[18] N. R. F. Braga, R. da Mata, Phys. Lett. B (2018)135918 [arXiv:2008.10457 [hep-th]].[19] P. Colangelo and F. Loparco, Phys. Lett. B (2019)500 [arXiv:1811.05272 [hep-ph]].[20] L. F. Ferreira and R. da Rocha, Phys. Rev. D (2020)106002 [arXiv:2004.04551 [hep-th]].[21] R. A. C. Correa, D. M. Dantas, C. A. S. Almeidaand R. da Rocha, Phys. Lett. B (2016) 358[arXiv:1601.00076 [hep-th]].[22] W. T. Cruz, D. M. Dantas, R. V. Maluf andC. A. S. Almeida, Annalen Phys. (2019) 1970035[1810.03991 [gr-qc]].[23] C. O. Lee, Phys. Lett. B (2020) 135030 [arXiv:1908.06074 [hep-th]].[24] D. Bazeia, D. C. Moreira and E. I. B. Rodrigues, J. Magn.Magn. Mater. (2019) 734.[25] A. Alves, A. G. Dias, R. da Silva, Physica (2015) 1[arXiv:1408.0827 [hep-ph]].[26] A. Alves, A. G. Dias, R. da Silva, Nucl. Phys. B (2020) 115137 [arXiv:2004.08407 [hep-ph]].[27] M. Gleiser and D. Sowinski, Phys. Rev. D (2018)056026 [arXiv:1807.07588 [hep-th]].[28] N. R. Braga, Phys. Lett. B (2019) 134919[arXiv:1907.05756 [hep-th]].[29] N. F. Braga, O. Junqueira, [arXiv:2010.00714 [hep-th]].[30] R. Casadio and R. da Rocha, Phys. Lett. B (2016)434 [arXiv:1610.01572 [hep-th]].[31] A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys.Rev. D (2006) 015005 [hep-ph/0602229].[32] C. Csaki, M. Reece, JHEP (2007) 062 [arXiv:hep-ph/0608266].[33] M. ´A. Mart´ın Contreras, A. Vega and S. Cort´es, Chin. J.Phys. (2020) 715 [arXiv:1811.10731 [hep-ph]].[34] N. R. F. Braga, M. A. Martin Contreras, S. Diles, Phys.Lett. B (2016) 203 [arXiv:1507.04708 [hep-th]].[35] E. Katz and M. D. Schwartz, JHEP (2007) 077[arXiv:0705.0534 [hep-ph]].[36] N. R. F. Braga, M. A. Martin Contreras, S. Diles, Eur.Phys. J. C (2016) 598 [arXiv:1604.08296 [hep-ph]].[37] P. Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeauand S. Nicotri, Phys. Rev. D (2008) 055009[arXiv:0807.1054 [hep-ph]].[38] E. Witten, Adv. Theor. Math. Phys. (1998) 253 [hep-th/9802150].[39] A. Vega and I. Schmidt, Phys. Rev. D (2009) 055003[arXiv:0811.4638 [hep-ph]].[40] H. Boschi-Filho, N. R. F. Braga, F. Jugeau andM. A. C. Torres, Eur. Phys. J. C (2013) 2540[arXiv:1208.2291 [hep-th]].[41] T. Branz, T. Gutsche, V. E. Lyubovitskij, I. Schmidtand A. Vega, Phys. Rev. D (2010), 074022[arXiv:1008.0268 [hep-ph]].[42] A. Vega, I. Schmidt, T. Branz, T. Gutsche andV. E. Lyubovitskij, Phys. Rev. D (2009) 055014[arXiv:0906.1220 [hep-ph]].[43] S. Cort´es, M. ´A. Mart´ın Contreras, J. R. Rold´an, Phys.Rev. D (2017) 106002 [arXiv:1706.09502 [hep-ph]].[44] M. Rinaldi and V. Vento, [arXiv:2101.02616 [hep-ph]].[45] C. Amsler et al., Eur. Phys. J. C (2004) 23.[46] A. Masoni, C. Cicalo, G. L. Usai, J. Phys. G (2006) R293.[47] S. He, M. Huang and Q. S. Yan, Phys. Rev. D (2010)014003 [arXiv:0903.5032 [hep-ph]].[48] S. Godfrey and J. Napolitano, Rev. Mod. Phys. (1999)1411 [arXiv:hep-ph/9811410 [hep-ph]].[49] C. Amsler, N. A. Tornqvist, Phys. Rept. (2004) 61.[50] E. Klempt and A. Zaitsev, Phys. Rept. (2007) 1[arXiv:0708.4016 [hep-ph]].[51] D. V. Bugg, L. Y. Dong and B. S. Zou, Phys. Lett. B458