Non-linear Regge trajectories with AdS/QCD
NNon-linear Regge trajectories with AdS/QCD
Miguel Angel Martin Contreras ∗ and Alfredo Vega † Instituto de F´ısica y Astronom´ıa,Universidad de Valpara´ıso,A. Gran Breta˜na 1111, Valpara´ıso, Chile (Dated: April 23, 2020)In this work, we consider a non-quadratic dilaton Φ( z ) = ( κ z ) − α in the context of the staticsoft wall model to describe the mass spectrum of a wide range of vector mesons from the light upto the heavy sectors. The effect of this non-quadratic approach is translated into non-linear Reggetrajectories with the generic form M = a ( n + b ) ν . We apply this sort of fits for the isovectorstates of ω , φ , J/ψ and Υ mesons and compare with the corresponding holographic duals. We alsoextend these ideas to the heavy-light sector by using the isovector set of parameters to extrapolatethe proper values of κ and α through the average constituent mass ¯ m for each mesonic specieconsidered. In the same direction, we address the description of possible non- q ¯ q candidates using ¯ m as a holographic threshold, associated with the structure of the exotic state, to define the values of κ and α . We study the π mesons in the light sector, and the Z c , Y and Z b mesons in the heavy sectoras possible exotic vector states. Finally, the RMS error for describing these twenty-seven states withfifteen parameters (four values for κ and α respectively and seven values for ¯ m ) is 12 . I. INTRODUCTION
Nowadays, there is no doubt that hadrons are boundstates of quarks and gluons, whose interactions are de-scribed by quantum chromodynamics (QCD). This quan-tum field theory is endowed with a coupling constant thatcontrols the energy of the hadronic processes. At high en-ergies, the smallness of the coupling constant makes thetheory perturbative. On the other hand, the low energybehavior is non-perturbative. It is precisely in the latterregime where several hadronic properties are found. Also,the developed perturbative theoretical tools are insuffi-cient to describe this particular hadronic physics. Thisissue motivated the development of techniques and toolsthat allows the direct use of QCD in the study of hadrons,such as Lattice QCD (e.g.[1]) or the use of the DysonSchwinger equations to study hadrons (e.g.[2]).This picture has also prompted the development ofphenomenological models inspired by QCD, capturingimportant properties of the interaction between quarksand gluons, offering us alternatives to perform calcula-tions of hadronic properties.A successful example of phenomenological models forthe study of hadrons is the so-called quark potential mod-els [3–7], which have been remained valid since the mid-dle seventies when the first heavy quark mesons, the
J/ψ meson was observed. In this approach, the Schrdingerequation, with a potential describing the interaction be-tween constituent heavy quarks inside the meson, pro-vides good results describing the mesonic spectra andother properties related to the hadronic wave function,such as the decay constants [8].From the QCD point of view, it is possible to infer the ∗ Electronic address: [email protected] † Electronic address: [email protected] behavior of the potential when the constituent quarks areclose or far between them. In the former case, the large Q limit, the coupling constant is small enough, allow-ing to use perturbative techniques to describe the quarkinteraction by considering the one-gluon exchange only.As a result, the potential is found to be Coulomb-likein this limit. On the other case, In the long-inter-quarkdistance. or small Q limit, the strong coupling constantbecomes large, preventing any perturbative machinery.In this case, quarks are considered as confined partons .This part of the potential cannot be explored by analyti-cal QFT methods. But, extensive developments in latticeQCD proved that this term seems to be linear [9]. Simi-lar results were found on the holographic side, where thedictionary establishes that a closed string world-sheet isdual to the Wilson loop, which accounts for confinementon the boundary theory [10–12].Summarizing, today we know that the constituentquark interaction potential is well-motivated from QCD:it must interpolate between a Coulomb-like potential atshort distances and a linear-like potential at long dis-tances. In order to fit this phenomenological suggestion,several alternatives have been proposed (See [3–5]). Thesimplest realization of these sorts of ideas, giving excel-lent results, is just the sum of both contributions.Another succesfull possibility of building phenomeno-logical models is to study hadron properties using gauge/gravity correspondence . Namely the so-calledbottom-up AdS/QCD approach allows us to calculatehadronic properties by capturing the main strong inter-action features of hadrons in different mediums in an5-dimensional AdS-like metric tensor and other back-ground fields, as the dilaton.In bottom-up kind of models, a dilaton field is usedto induce confinement on the dual boundary theory. If a r X i v : . [ h e p - ph ] A p r Isovector meson masses (PDG) n ω (MeV) φ (MeV) ψ (MeV) Υ (MeV) . ± .
12 1019 . ± .
016 3096 . ± .
011 9460 . ± .
262 1 . − ±
20 3686 . ± .
012 10023 . ± .
323 1670 ±
30 2135 ± ± ± ± .
54 1960 ± − − − ± . ± .
25 2290 ± − − − − − − . +3 . − . − − − − − − − − − . +10 . − . Table I: This table summarises the experimental masses ([13]) for isovector mesons families consisting on ω , φ , ψ and Υ radialstates. the dilaton considered is static and quadratic [14], con-finement is manifest by the appearance of linear Reggetrajectories in the mesonic sector. Further works con-sider other possible forms of the dilaton field that inter-polates the quadratic dilaton at high z , keeping lineartrajectories (for higher quantum numbers) and allowingthe study of other phenomena as chiral symmetry break-ing. Linear Regge trajectories are a good description ofthe mesonic mass spectra in the light sector, and this hasbeen used traditionally as a guideline in order to catchhadron properties in the AdS side of AdS / QCD models.But if hadrons contain s or heavy quarks, linearity in tra-jectory is lost [15–19]. For this reason, we explore otherkind of dilatons in order to describe hadrons where lin-ear Regge trajectories disagree with experimental data.This could be interesting at moment to study, for exam-ple, heavy mesons in holographic models, because as youcan see in literature [20–24], AdS/QCD models appliedto charmonium or bottomonium spectra are no so goodenough to describe them, despite the fact that other ob-servables (as the melting temperature) have the properqualitative behaviour.This work has been structured as follows: in sec-tion II we consider four families of isovector mesonswith different constituent quarks and we show that thesemass spectra agree with a non-linear Regge trajectory,parametrized by M = a ( n + b ) ν . We associate the index ν with the average constituent quark mass in each case,and then we propose an expression for this index.In section III we review holographic recipe to describemesonic masses. We conjecture that the linearity de-viation in Regge trajectories is associated, in the AdSside, with changes in the background fields, namely, thedilaton profile. We propose a dilaton deformation of theform z − α , where α encodes the effect of the averageconstituent quark mass on the Regge trajectory. In sec-tion IV, we apply these ideas to the description of radialisovector states ( ω , φ , J /ψ and Υ), with quantum num-bers defined as I G J P C = 0 − (1 −− ).In section In section V we used the isovector non-lineartrajectories fitted to extrapolate the values of κ and α for K ∗ and the heavy-light vector mesons. We also make a description of non- q ¯ q states by testing at the holo-graphic level some of the proposals to describe exoticmesons as multiquark states or gluonic excitations. Thisexotic states can be described by considering the confor-mal dimension ∆ associated with the operator that cre-ates these states and how ∆ affects the bulk mass termin the associated holographic potential. In this case, weuse ¯ m as a holographic threshold, defined in terms of thestructure of each exotic state, to define the values of κ and α . We consider exotic candidates in the light sector( π meson) as well as in the heavy one ( Z c , Z b and X mesons).Finally, in section VI we expose the conclusions andfinal comments about the present work. II. NON-LINEAR TRAJECTORIES
The relation between hadronic squared mass and ra-dial (and orbital) quantum number is considered usuallyas linear. This affirmation is in general, accepted dueto experimental evidence, is especially true in the lightsector, but when quark masses are increased a non-linearRegge trajectory seems better to describe hadron spectra[15–19].In this work we consider four families of isovectormesons labeled as I G J P C = 0 − (1 −− ), and investigatelinear and non-linear expressions for M . In table I wesummarize the experimental masses of all four isovectormeson families considered in our analysis. In table II weshow our fits for a linear and non-linear Regge trajectoryexpressed as M n = a ( n + b ) . (1)and M n = a ( n + b ) ν (2)In Fig. 1, we summarize experimental data using bothfits. Linear Regge Trajectory: M = a ( n + b ) Non Linear Regge Trajectory ( M = a ( n + b ) ν )Meson a b R a b ν R ω . − . . . − . . . φ . − . . . − . . . ψ . . . . . . . . . . . . . . R . Observe that linear fits bring good descriptionof the trajectories, but R decrease from unity when we increase the quark constituent mass. Also notice that the non-linearfit is more precise since R is bigger than the linear one in each case. M ω M ϕ M ψ M Υ Figure 1: This plot shows M vs n for different vector mesons ( ω, φ, ψ and Υs). Dots represent experimental data, and in eachpanel there are two continuous lines, one represent the best linear fit ( M = a ( n + b )) and the other corresponding to a nonlinear fit ( M = a ( n + b ) ν ). As it is possible to see from the analysis of R in tableII and plot 1, the non-linear fit describe better (more ac-curate) the relationship between M and n . Notice thatthe exponent ν used in the non-linear case decreases whenthe constituent quark mass is higher. We suggest that ν is a function of the average constituent quark masses thatcompose the hadron. We also propose an expression tofit the four exponents ν appearing in table II plus anadditional point suggested by the chiral limit, i.e., addi-tionally, we consider ν = 1 when constituent quarks aremassless.The average constituent quark masses mentioned above can be defined as¯ m ( q , q ) = 12 ( m q + m q ) . For the constituent quark masses, we use the followingset of values m u = 0 .
336 GeV , m d = 0 .
340 GeV , m s = 0 .
486 GeV m c = 1 .
550 GeV , m b = 4 .
730 GeV ω with α = 0 . and κ = 498 MeV φ with α = 0 . and κ = 585 MeV n M Exp (MeV) M Th (MeV) R. E. (%) n M Exp (MeV) M Th (MeV) R. E. (%) . ± .
12 981 .
43 25 . . ± .
016 1139 .
43 11 .
82 1400 − . ±
20 1583 5 .
83 1670 ±
30 1674 0 .
25 3 2135 ± ± ±
25 1967 1 . − − ±
20 2149 6 . − − M = 0 . .
012 + n ) . with R = 0 . M = 1 . . n ) . with R = 0 . ψ with α = 0 . and κ = 2150 MeV Υ with α = 0 . and κ = 11209 MeV n M Exp (MeV) M Th (MeV) R. E. (%) n M Exp (MeV) M Th (MeV) R. E. (%) . ± .
011 3077 .
09 0 .
61 1 9460 . ± .
26 9438 . .
232 3686 . ± .
012 3689 .
62 0 . . ± .
32 9923 .
32 0 .
783 4039 ± . .
44 3 10355 ± . . .
754 4421 ± . .
77 4 10579 . ± . . .
195 Not Seen − − . +3 . − . . .
886 Not Seen − − . +10 . − . . . M = 8 . .
287 + n ) . with R = 0 . M = 76 . .
901 + n ) . with R = 0 . . The last column on each set of data is the relative error per state. Experimentalresults are read from PDG [13]. For the exponent ν introduced in the non-linear fit (2),we propose the following parameterization in terms of theaverage constituent quark mass given ν = a ν + b ν e ( − c ν ¯ m ) , (3)with the following model parameters a ν = 0 . , b ν = 0 . , c ν = − . . Notice that in the massless constituent quark case, i.e.¯ m = 0, we recover linearity. m α Figure 2: This plot shows the behavior of the dilaton expo-nent α as a function of the quark constituent mass. Noticethat for low masses, dilaton should be quadratic, implying theappearance of linear Regge trajectories for such states. III. GEOMETRIC BACKGROUND
Let us consider a five-dimensional AdS Poincar patchdefined by the following metric dS = e A ( z ) (cid:2) dz + η µν dx µ dx ν (cid:3) , (4)Also, we consider a bulk vector field A m ( z, x ) dual toisovector mesonic states interacting with a static dila-ton Φ( z ). This dilaton is motivated by the particle phe-nomenology to model confinement through the appear- m ν Figure 3: ν exponent as a function of the average constituentquark mass. For the massless case, it should be expected torecover ν = 1. K ∗ with α = 0 . and κ = 531 .
24 MeV n M Exp (MeV) M Th (MeV) R. E. (%) . ± . . .
22 1414 ±
15 1451 . .
63 1718 ±
18 1754 . . K ∗ radial mesonic states. The last column is the relative error per state.Experimental results are read from PDG [13]. ance of Regge trajectories. The action for such fields is I V = − g (cid:90) d x √− g e − Φ( z ) F mn F mn . (5)We have assumed the bulk vector field as massless sincefor mesons this quantity is fixed to be zero.From this action, and imposing the gauge fixing A z =0, we arrive at the following equation of motion for thebulk field ∂ z (cid:104) e − B ( z ) ∂ z A µ ( z, q ) (cid:105) + ( − q ) e − B ( z ) A µ ( z, q ) = 0 , (6)where we have introduced B ( z ) = Φ( z ) − A ( z ). Let usspan the bulk vector field as A µ ( z, q ) = A µ ( q ) ψ ( z, q )in order to transform the equation of motion into aSchrdinger-like one. Performing the Boguliobov trans-formation ψ ( z ) = e Φ( z ) / φ ( z, q ) we arrive to the the fol-lowing expression − φ (cid:48)(cid:48) ( z, q ) + V ( z ) φ ( z, q ) = ( − q ) φ ( z, q ) (7)where the holographic potential V ( z ) is defined as V ( z ) = 14 B (cid:48) ( z ) − B (cid:48)(cid:48) ( z ) (8)= 34 z + Φ (cid:48) ( z )2 z + Φ (cid:48) ( z ) − Φ (cid:48)(cid:48) ( z )2 , (9)where we have used the warp factor A ( z ) = log( R/z ).The hadronic mass spectra, and the Regge trajecto-ries, are constructed from the eigenvalues of this poten-tial, which is fixed by the structure of the B ( z ) function.In the context of the original soft wall model [14], thepotential is fixed by B ( z ) = κ z − log( R/z ) obtainingthe linear spectrum M n = 4 κ ( n + 1) , (10)associated with vector mesons with massless constituentquarks. The Regge slope is identified to the κ , in units ofGeV, that fixes the scale of the trajectory. The linearityobserved in (10) is achieved by the specific quadratic formof the dilaton, which induces a z behavior at high- z in m κ Figure 4: This plot shows how the dilaton scale κ runs withthe average constituent mass. For the massless case we haveused κ = 0 .
388 GeV [14]. the confining potential, that is translated into the lineardependence with the radial excitation number n .If we want to consider the case when the constituentquarks are massive, we must extend to non-linear Reggetrajectories [15–19]. Here we consider which this non-linearity is connected with the quark constituent mass.Therefore, massless quarks are tied to linear trajectories.Beyond the chiral symmetry breaking, once the quarksget mass, the trajectory ceases to be linear but remainsas a good approximation in the light sector. A deviationin linearity in the spectrum must be associated with achange from the usual quadratic static dilaton. This de-viation associated with the constituent quark mass is as-sociated can be parametrized into a shift in the quadraticexponent of the dilatonΦ( z ) = ( κ z ) − α . (11)Fixing α to be zero, we have the massless constituentquarks. In the following sections, we will discuss themassive constituent quarks case. Exponent for dilaton(this fit consider an additional point suggested by chirallimit) α = a α − b α e ( − c α ¯ m ) , (12)with the following set of parameters: State I ( J P ) q q κ (MeV) α M Exp (MeV) M Th (MeV) R. E. (%) K ∗ (782) 1 / − ) d ¯ s .
24 0 .
055 895 . ± . . . D ∗ (2007) 1 / − ) c ¯ u . ± .
05 1902 . . D +0 (2010) 1 / − ) c ¯ d . .
262 2010 . ± .
05 1906 . . D ∗ + s ? ) c ¯ s . .
296 2112 . ± . . . B ∗ + / − ) u ¯ b . .
800 5324 . ± .
22 4561 . . B ∗ / − ) d ¯ b . .
801 5324 . ± .
22 4564 . . B ∗ s − ) s ¯ b . .
809 5415 +1 . − . . . D ∗ + s has not been plenty identified, their decay modes areconsistent with J P = 1 − . See [13] for further details. a α = 0 . , b α = 0 . , c α = 0 . , and for the energy scale κ we have the following fit κ ( ¯ m ) = a κ − b κ e − c κ ¯ m , (13)with the following fit coefficients: a κ = 15 . , b κ = 14 . , c κ = 0 . IV. MESON MASSES
In AdS/QCD models, mesonic states are identified bythe bulk field mass dual to the hadronic states in con-sideration. This connection is made via the conformaldimension ∆ that fixes how the bulk field scales at theboundary. On the field theory side, the matching is re-alized by considering ∆ as the scaling dimension of theoperator that creates hadrons. In the case of mesons,∆ = 3.In general, from the standard AdS/CFT dictionary, wehave M R = (∆ − S )(∆ + S − , (14)where S stands for the hadron spin. In the case of isovec-tor mesons, ∆ = 3 and S = 1 implying that such bulkvector fields are massless as we considered in the sectionabove.Now let us turn our attention to the holographic poten-tial (8) constructed with the non-quadratic dilaton sug-gested in expression (11). This potential has the genericform V q ¯ q ( z, κ, α ) = 34 z − α κ ( κ z ) − α + 14 α κ ( κ z ) − α + 32 α κ ( κ z ) − α − κ ( κ z ) − α − α κ ( κ z ) − α + κ ( κ z ) − α + κz ( κ z ) − α − α κ z ( κ z ) − α . (15) Notice that the massless constituent quark case, when α = 0, we recover the potential for vector mesons [14].At this point, α becomes an extra parameter to beconsidered in the model. But, as we can see later, itis possible to parametrize a specific form depending on¯ m . This also will define a running for the slope κ . Toproperly construct these radial states, we will solve theSchrdinger equation associated with the potential (15)with α and κ as entries, for the chosen isovector family.Numerical results for each family are summarized in tableIII. V. EXTRAPOLATION TO OTHER MESONICSPECIES
Let us apply the ideas developed above to other vectormesonic systems at hand: Regge trajectory for vectorkaons and masses for vector heavy-light mesons. Thekey idea is to use the running of α and κ with the quarkconstituent mass as calibration curve to extrapolate theproper values for other mesonic samples. A. Kaons
Vector kaons are mesonic states labeled by I ( J P ) =1 / + ), with S = ± C = B = 0. In order to set thevalues for α and κ , we will use the following definition forthe quark constituent mass as the average of the massesof s and d quarks: ¯ m K ∗ = m s + m d . (16)With this mass, we found for the K ∗ system the fol-lowing values for κ and α : κ K ∗ = 531 .
24 GeV ,α K ∗ = 0 . . Results for this trajectory are summarized in table IV.
B. Heavy-light mesons
Heavy-light mesons are defined as hadronic systemswhere one of the constituent quarks is heavy (i.e., charmor bottom) whilst the other is light flavored (up, downor strange). The physics of these heavy-light hadronshas become one of the vastest fields of research in parti-cle physics. Calculations of the heavy-light mass spectraare done in the context of effective QFT [25], potentialmethods [26], QCD sum rules [27], Bethe-Salpeter equa-tion [28] and lattice QCD [29].Following the same procedure done in the case of vectorkaons, we can fix the quark constituent mass ¯ m as anaverage of the pair of constituent quarks inside the heavy-light meson. The mass spectrum and the correspondingvalues of κ and α are summarized in table V. C. Non- q ¯ q vector states All of the mesonic states with quantum numbers notallowed by the usual q ¯ q model are called exotic . A goodreview of the physics of such states can be found in [30–32] and references therein. We will focus on the vectorexotic states in this section. At holographic level, [33]addresses the exotic meson spectra for Z c and Z b in thecontext of Sakai-Sugimoto models.Holographically, as we explain above, the hadronicidentity is controlled by the scaling dimension associatedwith the operator that creates hadrons. This informationis encoded into the bulk mass of the five-dimensional fielduse to mimic hadrons. Equation (14) summarizes this.Therefore, if we identify the dimension of the operatorsthat create exotic states we can address the associatedvector mass spectrum by using the proper holographicpotential, that in our case has the specific form V non- q ¯ q ( z, κ, α, M , ∆ ) = V q ¯ q ( z, κ, α ) + M (∆) R z , (17)where V q ¯ q ( z, κ, α ) is given by the expression (15). Here,we will consider the exotic meson vector states organizedinto two groups: multi-quark states . and gluonic excita-tions . The former is associated with tetraquarks, hadro-quarkonium, and hadronic molecules. Pentaquarks arealso part of this category. For the sake of simplicity,we will devote to multi-quark states candidates with justfour constituent quarks. The methods developed herecan be extrapolated to multiquark candidates also.At this point, it is important to mention the gauge in-variance, since now we have massive bulk fields. Recallthat the gauge invariance should be manifest at the con-formal boundary, where all of the dual fields are massless[21]. The presence of the non-zero bulk mass does not af-fect the gauge A z = 0. If we pay attention to the massive e.o.m for the vector bulk fields, i.e., for the z component (cid:3) A z − ∂ z ( ∂ µ A µ ) + M e A A z = 0 , (18)and the spacetime components ∂ ν (cid:2) e − B ( ∂ µ A µ ) + ∂ z (cid:0) e − B A z (cid:1)(cid:3) − (cid:8) ∂ z (cid:2) e − B ∂ z A ν (cid:3) + e − B (cid:3) A ν − e − B e − A M A ν (cid:9) = 0 , (19)we can realize that the A z = 0 gauge condition still im-plies ∂ µ A µ = 0. Therefore, the fields at the boundaryare still transverse.The gluonic excitations cathegory classifies glueballsand hybrid mesons. In this paper we will focus on vectorhybrid mesons only, consisting of a quark-antiquark pairwith a finite number of active gluons.
1. Multi-quark states
In the case of multi-quark states, a degeneracy appearswhen the conformal dimension is defined. Furthermore,since the conformal dimension is counting indirectly thenumber of constituent quarks, this dimension does notdistinguish between four quarks in the diquark antidi-quark pair, the hadroquarkonium or hadronic moleculeconfigurations.This degeneracy can be removed if we consider theconstituent mass of each multi-quark configuration as acollection of N constituents, quarks or mesons, given by¯ m multi-quark = N (cid:88) i =1 ( P quark i ¯ m q i + P meson i m meson i ) , (20)with the condition that N (cid:88) i =1 ( P quark i + P meson i ) = 1 . (21)Notice that each weight P quark(meson) i measures thecontribution of a given constituent (quark or meson) withmass m quark(meson) . Each multi-quark state has a differ-ent mass configuration, used to calculated the parameters κ and α in the respective calibration curves, as we did inthe heavy-light mesons case. Diquark constituent model.
Tetraquark states canbe considered as hadronic states consisting of a pair of diquark and an anti-diquark interacting between them.A diquark is a non-colored singlet object used as es-sential building blocks forming tetraquark mesons andpentaquark baryons. These fundamental blocks either acolor anti-triplet or a color sextet in the SU(3) color rep-resentation [34]. These diquarks are bounded by spin-spin interactions. The constituent diquark approach is
Holographic spectrum Non- q ¯ q states ∆ = 6 and ¯ m diquark-antidiquark Multiquark state α = 0 . and κ = 2151 MeV I G ( J CP ) = 1 + (1 + − ) Z c mesonsn M Th (MeV) n State M Exp (MeV) ∆ M (%) . Z c (3900) 3887 . ± . .
02 4384 . Z c (4200) 4196 +35 − .
53 4706 . Z c (4430) 4478 +15 − .
1∆ = 6 and ¯ m hadronic molecule Multiquark state α = 0 . and κ = 2151 MeV I G ( J CP ) = 1 + (1 + − ) Z c mesonsn M Th (MeV) n State M Exp (MeV) ∆ M (%) . Z c (3900) 3887 . ± . .
822 4213 . Z c (4200) 4196 +35 − .
433 4551 . Z c (4430) 4478 +15 − .
64∆ = 6 and ¯ m Hadrocharmonium
Multiquark state α = 0 . and κ = 2523 MeV I G ( J CP ) = 0 + (1 −− ) Y or ψ mesonsn M Th (MeV) n State M Exp (MeV) ∆ M (%) . ψ (4260) 4230 ± .
252 4577 . ψ (4360) 4368 ±
13 4 .
83 4871 . ψ (4660) 4643 ± .
9∆ = 6 and ¯ m Hadronic Molecule
Multiquark state α = 0 . and κ = 1548 . MeV I G ( J CP ) = 0 + (1 −− ) Y or ψ mesonsn M Th (MeV) n State M Exp (MeV) ∆ M (%) . ψ (4260) 4230 ± .
372 4383 . ψ (4360) 4368 ±
13 0 .
353 4705 . ψ (4360) 4643 ± .
34∆ = 6 and ¯ m hadronic molecule Multiquark state α = 0 . and κ = 11649 MeV I G ( J CP ) = 1 + (1 + − ) Z B mesonsn M Th (MeV) n State M Exp (MeV) ∆ M (%) . Z B (10610) 10607 . ± .
852 10669 . Z B (10650) 10652 . ± . .
16∆ = 5 and ¯ m Hybrid Meson
Gluonic excitation state α = 0 . and κ = 488 MeV I G ( J CP ) = 0 − (1 + − ) π mesonsn M Th (MeV) n State M Exp (MeV) ∆ M (%) . π (1400) 1354 ±
25 0 .
162 1646 . π (1600) 1660 +15 − .
8∆ = 5 and ¯ m Hybrid meson
Gluonic Excitation α = 0 . and κ = 2151 MeV I G ( J CP ) = 1 + (1 + − ) Z c mesonsn M Th (MeV) n State M Exp (MeV) ∆ M (%) . Z c (3900) 3887 . ± . .
242 4156 . Z c (4200) 4196 +35 − .
943 4513 . Z c (4430) 4478 +15 − .
78∆ = 7 and ¯ m Hybrid Meson
Gluonic excitation state α = 0 . and κ = 11649 MeV I G ( J CP ) = 1 + (1 + − ) Z B mesonsn M Th (MeV) n State M Exp (MeV) ∆ M (%) . Z B (10610) 10607 . ± .
522 10696 . Z B (10650) 10652 . ± . . q ¯ q states considered in this work. Experimental results are read from PDG[13]. useful to describe the spectroscopy and decays of multi-quark states. It is expected that these diquark composedcandidates appear as poles in the S -matrix, described bynarrow widths.Theoretical approaches are done in the QCD sumrules [35, 36], potential models [37] framework, andlattice QCD [38], where they approach the diquark-antidiquark. Experimentally, charmonium and bottomo- nium tetraquark states can be identified because they de-cay into open-flavor states instead of a quarkonium witha light meson due to the spin-spin interaction dominance (See [30]).Following PDG [13], the charmonium Z c states, withquantum numbers I G ( J CP ) = 1 + (1 + − ), are candidatesto be vector tetraquarks. For these states we can use thecharm quark constituent mass given in section II to findthe values of κ and α for these states. Following Bam-brilla, we consider the Z c states as a single trajectory.Table VI summarizes the experimental candidates.Other studies, as [39], suggest ψ (4260) with 0 + (1 −− )as a vector tetraquark instead of Z c . As we will provelater, at least at the holographic level, ψ (4260) seems tobe consistent with the hypothesis that it is a hadrochar-monium state.In this case, we can organize the diquark and antidi-quark as ( q ¯ q ) ( q ¯ q ) , with conformal dimension ∆ = 6.This implies that the bulk mass is M R = 15 for thesestates. The parameter ¯ m can be set as a sort of holo-graphic threshold that will allow us to distinguish be-tween multiquark state descriptions.In the case of the diquark constituent model, thethreshold in this charmonium-like diquark-antidiquarkcase is set as ¯ m diquark-Antidiquark = ¯ m c , (22)implying that P quark i = 1 / P meson i = 0 for this con-figuration. With this definition we can set the propervalues for κ and α . Numerical results for this approxi-mation are shown in the first left panel of table VI. Hadroquarkonium model.
The hadroquarkoniumstates can be constructed by considering a vector me-son core with a cloud of two quarks [40]. From theexperiments, it was observed that most of the candi-dates to be heavy exotic states appear as final statescomposed by heavy quarkonium and light quarks. Thismotivated the idea that these states were made of a com-pact heavy quarkonium core surrounded by a light quarkcloud [41]. This quarkonium core interacts with the lightquark cloud through a colored Van der Waals-like force (similar as the one in molecular physics), allowing thedecay of these states into the observed quarkonium coreand the light quarks [42].Following [42], we will suppose that the states ψ (4260), ψ (4360) and ψ (4660) with 0 + (1 −− ) are possiblehadrocharmonium states, forming a single vector trajec-tory. Holographically, the operator that creates thesestates has dimension six, i.e., ∆ = 6, implying that thebulk mass is M R = 15, as in the case of the diquark-antidiquark pair configuration. The difference will be thedefinition of the holographic threshold used to set the pa-rameters α and κ .In this case, we will consider a charmonium ( J/ψ me-son) core characterized by its mass plus the light quarkcloud, consisting of a pair of u and d quarks. Therefore,the holographic threshold is defined as¯ m Hadrocharmonium = 12 m J/ψ + 14 ( ¯ m u + ¯ m d ) . (23)With this criterion, we can extrapolate the model pa-rameter and compute the mass spectra for these exotic states. The summary of these results is shown in thethird left panel of the table VI. Another possible can-didates to be vector hadrocharmonium are the pair ofstates χ c (3872) and χ c (4140) [43], with quantum num-bers given by 0 + (1 ++ ). In our case, the model devel-oped here is not sensitive to such difference between thequantum numbers, i.e., the transition CP = ++ → −− is not described by this non-quadratic dilaton (11), im-plying that for us these states are degenerate. A simi-lar situation occurs in other bottom-up models, such as[44], where is not possible to distinguish between mesonicstates with different isospin since the model does not con-sider chiral symmetry breaking. In our case, we need toadd extra parameters to split up these two sets of exoticstates. Hadronic molecule model. hadronic molecules arestates conformed by a pair of internal mesons boundedby strong QCD forces, interacting between them via aresidual weak QCD colorless force [32]. These structurescan be realized as a two heavy quarkonia interacting orone heavy quarkonium plus a light meson. This pro-posal is as old as QCD itself [45]. The first theoreticalapproaches are applications of the deuteron Weinberg’smodel [46, 47]. Experimental results for X (3872) and D S (2317) are consistent with these ideas. Other ap-proaches are done in the context of sum rules [48] orlattice QCD [49].In the heavy sector, [30] and [31] suggest that Z c or Y mesons could be possible hadronic molecule charmoniumstates, containing at least one pair of c ¯ c in the inner coreof the molecule. The most relevant decay of these statesis J/ψ π . Following this, we will construct the thresholdmass for the holographic Y or ψ mesons as¯ m hadronic molecule = 13 m J/ψ + 23 m ρ . (24)In the case of the Z c mesons, we have proposed thefollowing threshold mass¯ m hadronic molecule = 0 . m J/ψ + 0 . m ρ . (25)We will extend these ideas to the bottomoniumhadronic molecule candidates, the z B mesons, wherethe expected core is the Υ(1 S ) state. The holographicthreshold in this case is¯ m hadronic molecule = 0 . m Υ(1 S ) + 0 . m ρ . (26)Results for all of these fits are showed in the table VI.As in the other multiquark cases, the conformal dimen-sion is ∆ = 6, implying a bulk mass given by M R = 15.At this point, we can notice that, at the holographiclevel, Z c and Y states are better described as hadronicmolecules. When Z c is described as a pair of diquark-antidiquark, the RMS error (7.5%) is bigger than in thehadronic molecule case (2.5%). For the Y mesons we ob-serve the same: the RMS error in the hadrocharmonium0 Vector hybrid meson P q P ¯ q P G π .
497 0 .
497 6 × − Z c .
49 0 .
49 0 . Z b .
495 0 .
495 0 . Z b we are considering two flux tubesinstead one. description (6.8%) is bigger than in the molecular case(5.5%).
2. Gluonic excitations: Hybrid mesons
Gluonic excitations are defined as hadrons with con-stituent gluonic fields. QCD confined states are natu-rally non-perturbative, therefore it is not surprising tohave constituent gluons inside hadrons. This kind ofstructure is realized as pairs of quarks and anti-quarksjoined by gluonic flux tubes. This particular configura-tion allows us to introduce other sets of quantum num-bers not possible in the quark constituent model, for ex-ample, the J CP = 1 + − configuration that we will ex-plore in this section. The mesonic states consisting ofvalence quarks and constituent gluons are called hybridmesons . Another set of gluonic excitations are the glue-balls, not addressed here, characterized by the absenceof quark quantum numbers. In general, hybrid mesonshave been studied using flux tube model [50], the MITbag model [51], coulomb-like potentials [52], gluon con-stituent model [53], quenched QCD [54] or lattice QCD[55].Experimentally speaking, it is possible to find candi-dates across the entire mass range, from light mesons upto bottomonium states. In particular, we will focus onthe π , Z c and Z b states.To build up the holographic description, we need todefine the hadronic operators creating hybrid mesons.Following the standard AdS/QFT dictionary, the phe-nomenological motivation comes from the two-point func-tions at the conformal boundary. These objects are de-fined in terms of operators that are composites of quarksand gluons, that generally, can be defined as q γ µ ¯ q G µν ,where G is a gluonic field on its ground stated and γ µ are the Dirac matrices [56]. This, in terms of the op-erator dimension, means that ∆ = 5 if we consider onesingle constituent gluon, or ∆ = 7 if we consider twoconstituent gluons. This information is translated in thebulk mass as M R = 8 and M R = 24 respectively.Since we want to define the holographic threshold, weneed to infer a mass for the constituent gluon. Following[53], we adopt M G = 700 MeV. Therefore, our generalproposal for the holographic threshold has the form¯ m hybrid meson = P q m q + P ¯ q m ¯ q + P G m G . (27)In table VII we summarize the choices for the P i co-efficients used to describe the hybrid meson candidates in the context of the model developed here. With theseholographic thresholds, we can obtain the proper valuesfor κ and α to fix the non-linear trajectory. Results aredepicted in the last three panels of the table VI.The light and the charmonium were fitted supposing asingle constituent gluon, which is translated in a confor-mal dimension fixed as ∆ = 6. The RMS error in bothcases is about 1% in the former and 4 .
4% in the latter.In the case of the bottomonium hybrids, the best fitswere obtained for two constituent gluons, implying that∆ = 7. The RMS error, in this case, was near to 2 . Z c mesons are better described as a holographichadronic molecule (R.M.S near to 2.48 %) than a holo-graphic hybrid meson (R.M.S. near to 4.41 %). VI. DISCUSSIONS AND CONCLUSIONS
In the model AdS/QCD with dilaton, the usual ap-proach considers quadratic dilatons at large z , becauseconfiguration produces linear Regge trajectories. But itis important to notice that this sort of Regge trajecto-ries is a good description only in the light sector. Forthis reason we propose a new shape for dilaton field,(namely φ ( z ) = ( κz ) − α ), breaking the conformal invari-ance and producing trajectories with the generic form M n = a ( n + b ) ν . This set of trajectories reproduce, ina satisfactory form, masses for vector mesons with dif-ferent constituent quarks, catching linearity in the mass-less quark case and exhibiting how this linearity starts tocease when constituent quark masses increased.We consider that α and ν depend on the average ofconstituent quark masses for the mesons considered. Alsowe proposed a explicit shape for α ( ¯ m ) and ν ( ¯ m ) in orderto built a model that produce a good spectrum for vectormesons with different constituents.Nowadays in literature, it is possible to found somemodels AdS/QCD applied to charmonium or bottomo-nium [20–24], but spectra are no so good enough in thesemodels, although other observables (as the melting tem-perature) have the proper qualitative behavior. There-fore, these ideas, as we discussed here, can be useful inthis kind of application.1Regarding the chiral symmetry, even though the modeldescribes the spectra for φ and ω mesons, it does not re-produce a proper chiral symmetry breaking picture, asmost of the static soft wall-like models developed. Themain drawback is the impossibility to distinguish be-tween the explicit and the spontaneous breaking sincethe quark condensate σ q and the quark mass m q arenot independent. In the case of the dilaton proposedhere, although its UV behavior is different from the staticquadratic one, this does not guarantee the independencebetween m q and σ q . The best advances are done in theframe of dynamical AdS/QCD models, as [57, 58], or di-rect modifications of the bulk vev by changing the bulkmass, as it was done in [59].In this direction, linear trajectories are associated withlow constituent quark mass, as the results in tables II andIII are demonstrating. Therefore, we conclude that thesoft wall model is set before the chiral symmetry break-ing scenario. Furthermore, the meson spectra obtainedis degenerate: there is no form to distinguish ρ , ω and a vector mesons the using quadratic dilaton only. It isnecessary to do explicitly the chiral symmetry breakingby using SU(2) bulk vector fields and a tachyonic vev toaddress this, even though soft wall-like models do notrepresent a QCD-like chiral symmetry breaking. See forexample the analisys done in [60].By introducing the ν exponent in the radial Regge tra-jectory we can explore the effect of the quark mass. Asit was pointed out in [15–17], increasing the quark massshould deviate the trajectory from the linear case. Inthis holographic approach, such behavior was observed.Therefore, despite the fact we do not deal with the me-son structure directly, we can capture information aboutit in the non-linearity behavior. Translated to the bulkside, this information is captured in the α parameter,which measures the deviation from the quadratic formin the dilaton. As we observed in graphic 3, increasingconstituent quark mass implies a strong deviation fromthe quadratic dilaton.In the case of the energy scale κ , it is important to no-tice that its value is near to the constituent quark massfor each meson considered. Furthermore, in linear softwall model, κ defines the vector Regge slope (string ten-sion), i.e, 4 κ = a , where the linear trajectory is definedas M = a ( n + b ). In the holographic non-linear case,where the trajectory is defined as M = a ( n + b ) ν , theenergy scale κ also increases with the quark mass, in-dicating that it is connected indirectly with the mesonstructure. In tables III, IV and V we see that each κ increases with the constituent quark mass. In the case ofthe non-linear trajectory, a should be proportional to κ ,and also carries information about the string tension andthe quark constituent mass in each mesonic family. Re-call that in this case, the meson is modeled as the usual flux tube with two massive quarks at the ends, thus it isexpected that the quark mass information should appearin the slope. Thus, the factor a in the non-linear trajec-tory should be a function of α and ν . Moreover, from thedata reported in table III we can infer the fitted form for a as: a ( κ , α, ν ) = (11 . e − . α − . e − . ν ) κ , (28)where the correlation coefficient for this fit is R = 1.Notice that in the case where α = 0 (implying ν = 0) and κ = 388 MeV, we obtain a = 0 . κ , α and ν .It is worthy to say that this dilaton is not capturingthe expected low z behavior in the eigenmodes. Noticethat the meson ground states are not well fitted in thelight sector as long as the ν exponent. This can be in-ferred by the fact that trajectories are not exactly fitted.Therefore, this proposed dilaton should be interpolatedwith other low z proposals, as [23, 61]. But, on the otherhand, it was possible to fit heavy light vector mesons andto test, at holographic level, possibles candidates to benon- q ¯ q states, just by considering how the hadronic oper-ators at the boundary change their conformal dimension,that has information about the meson constituent indi-rectly, altogether with the holographic threshold ¯ m , thatparameterizes the structure of the state at hand. Thechange in the conformal dimension is translated into amodification of the bulk mass term that appears in theholographic potential (17), whilst ¯ m fixes in the calibra-tion curves (13) and (12) the values for κ and α . Thesame methodology was used to do the holographic fit forthe heavy-light mesons and the K ∗ vector states.As a final comment is important to recall the precisionof the holographic picture discussed in this manuscript.Although the ground states for light mesons were notwell fitted (errors near to 20%), the RMS for the modelwith 27 mesonic states (we do not consider the non- q ¯ q states since they are holographic predictions) with 15 pa-rameters (four values for κ and α respectively, and seven¯ m ), we get an error near to 12 . Acknowledgments
We wish to acknowledge the financial support providedby FONDECYT (Chile) under Grants No. 1180753 (A.V.) and No. 3180592 (M. A. M. C.). [1] S. Aoki et al., Eur. Phys. J.
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