PHYSICAL REVIEW D 101, 106002 (2020)

Nucleons and higher resonances: An AdS/QCD configurational entropic incursion

† Luiz F. Ferreira1,2,* and Roldao da Rocha 1, 1Federal University of ABC, Center of Mathematics, 09580-210 Santo Andr´e, Brazil 2Federal University of ABC, Center of Physics, 09580-210 Santo Andr´e, Brazil

(Received 10 April 2020; accepted 22 April 2020; published 1 May 2020)

P 1þ P 3− Families of J ¼ 2 (Nð939Þ, Nð1440Þ, Nð1710Þ, Nð1880Þ, Nð2100Þ, Nð2300Þ) and J ¼ 2 nucleons (Nð1520Þ, Nð1700Þ, Nð1875Þ, and Nð2120Þ) are scrutinized from the point of view of the configurational entropy (CE). The mass spectra of higher JP resonances in each one of these families are then obtained when configurational-entropic Regge trajectories, that relate the CE of families to both their JP spin and also to their experimental mass spectra, are interpolated. The mass spectra of the next generation of nucleon resonances are then compared to already established baryonic states in data group (PDG). Besides the zero temperature case, the finite temperature analysis is also implemented.

DOI: 10.1103/PhysRevD.101.106002

I. INTRODUCTION In what concerns (QCD) as a The configurational entropy (CE) represents a measure reliable description of and (and their inter- of spatial correlations, in the very same sense of the actions), and other elementary as well, their pioneering Shannon’s information entropy. The fundamen- intrinsic dynamical features make a difficult task of great tal interpretation of the CE consists of the limit to a lossless complexity to investigate hadronic nuclear states and compression rate of information that is inherent to any resonances. A particular approach to study nonperturbative physical system. The CE regards the minimum amount aspects of QCD is the AdS/QCD model. It has been very of bits necessary to encode a message [1]. There is no successful to provide general properties of hadronic states, code that takes less number of bits per symbol, on average, hadronic mass spectra, and chiral symmetry breaking. than the information entropy source. Codes closer to the These bottom-up models are implemented by considering information entropy are called optimal codes (or compres- deformations in the anti–de Sitter (AdS) space. For in- sion algorithms, in the case of noiseless sources), whose stance, the hard wall model consists in introducing an limit is the information entropy. This is the Shannon’s infrared (IR) cutoff corresponding to the confinement scale coding-source theorem. A physical system has maximal [7]. On the other hand, in the case of the soft wall model is entropy when the distribution of probability is uniform, included a field acting as a smooth IR cutoff. since this is the highest unpredictable case. The CE is a In both these regimes, the CE provides new procedures measure of information concerning the spatial complexity to figure out and unravel distinct physical phenomena. The of a system [2,3], meaning the possible correlations among CE also consists of a powerful tool to probe, predict, and its parts along the spatial domain. The CE has been applied corroborate to physical observables, including experimen- to nonlinear scalar field models with spatially localized tal and theoretical aspects. In fact, the CE has been playing energy density [4,5]. Modes in a system corresponding important roles in scrutinizing AdS/QCD (both hard wall to physical states with lower CE have been shown to be and soft wall) models, studying relevant properties in QCD more dominant and configurationally stable, being more and its phenomenology. The CE and the newly introduced detectable and observable, from the experimental point of concept of entropic Regge trajectories were used to inves- view [6]. tigate and predict the mass spectra of the next generation of higher spin particles of four light-flavor families *[email protected] [8,9], whereas an analog method was implemented in † [email protected] Ref. [10], to predict the mass spectra of higher spin tensor meson resonances, matching possible candidates in PDG Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. [11]. Previously, other aspects of the CE and mesonic states Further distribution of this work must maintain attribution to have been already studied in Ref. [6]. In addition, the the author(s) and the published article’s title, journal citation, interplay between CE and the stability of scalar in and DOI. Funded by SCOAP3. AdS/QCD was implemented in Ref. [12]. Reference [13]

2470-0010=2020=101(10)=106002(9) 106002-1 Published by the American Physical Society LUIZ F. FERREIRA and ROLDAO DA ROCHA PHYS. REV. D 101, 106002 (2020) paved the scrutiny of the bottomonium and charmonium devoted to a further analysis, discussion, and compilation production in the context of the CE in AdS/QCD. of the main results and their physical consequences. Phenomenological data show relative abundance and dominance of certain states, whose intrinsic II. HOLOGRAPHIC ADS/QCD MODEL information is more compressed in Shannon’s sense. Also, AND in the finite temperature case, it identifies higher phenom- P ¼ 1þ enological prevalence of lower S-wave resonances and The description of N baryons with spin J 2 in the lower masses quarkonia in nature [14]. References [15,16] AdS/QCD soft wall model can be accomplished by using – utilized the CE to probe the dependence on the impact different approaches [39 47]. We choose the method parameter of resonances production in qq¯ scattering, inside employed in Refs. [47,48] to describe , both at the color-glass condensate (CGC) model. Besides, the zero and finite temperature, in the AdS/QCD soft wall production of diffractive mesonic resonances in AdS/ model. Due the fact that the solutions in Refs. [47,48] are P QCD was proposed in Ref. [17], also in the CGC context. degenerate, for any value of half-integer J , and hold for l The CE was also employed to compute the cross composed of angular orbital momentum and section, in the holographic light-front wave function setup radial quantum number n, these solutions will be also P ¼ 3 − [18], whereas energy correlations in - applied for the N baryon family with J 2 . annihilation were studied in [19,20]. The anomalous To start, one considers the AdS-Schwarzschild metric in dimension of the target distribution and the nuclear conformal coordinates, CE were also shown to drive points of stability in particle 2 2 collisions in LHC [21]. The CE setup to heavy-ion 2 ¼ R ð Þ 2 − ⃗2 − dz ð Þ ds 2 fT z dt dx ; 1 collisions was deployed in Ref. [22]. Baryons were inves- z fTðzÞ tigated under the CE apparatus, where were ð Þ¼1 − 4 4 shown to have higher CE when compared to standard with fT z z =zh, where zh is proportional to the baryons [23]. Reference [24] studied quarkonium in finite inverse of the AdS-Schwarzschild black brane temperature density plasmas via the CE. as T ¼ 1=ðπzhÞ. Besides the quadratic dilaton field ϕðzÞ¼ Not only the AdS/QCD regime of AdS=CFT has been k2z2 usually employed in the AdS-QCD soft wall model, a −λ ð Þ explored from the point of view of the CE. Stellar thermal prefactor, e T z , is also introduced, where configurations and black holes were also explored via the CE [25]. References [26,27], besides showing that z2 z4 z6 λ ðzÞ¼α þ γ þ ξ ; ð2Þ Hawking-Page phase transitions can be driven by the CE, T z2 z4 z6 demonstrated that the bigger the black holes, the more h h h stable they are [28,29]. The CE plays an important role on where the dimensionless parameters γ and ξ have been studying Bose-Einstein condensates, as quantum fixed in Ref. [48] to guarantee gauge invariance and portrait models of black holes. In fact, the Chandrasekhar massless ground-state pseudoscalar (π, K, η)in collapse critical density does correspond to a CE critical the chiral limit, as well as to suppress the sixth power of the point [30,31]. Besides, the CE was shown to be an radial coordinate in the holographic potential. Precisely, appropriate paradigm to study phase transitions as CE JðJ−3Þþ3 2 2 2 γ ¼ and ξ ¼ . The term αz =z is a small critical points [4,5,25,32,33]. Topological defects were also 5 5 h perturbative correction to the dilaton ϕ, where the param- studied in Refs. [34–36]. The CE was also employed to eter α is related with the restoration of chiral symmetry at derive the Higgs and the masses in effective critical temperature. It is useful to relate the Regge-Wheeler theories [37,38]. tortoise coordinate, r, to the holographic coordinate, z, via This paper is organized as follows: Sec. II briefly the substitution introduces the soft wall AdS/QCD model, presenting the Z nucleon resonances and the obtention of their mass spectra. dz z 1 1 − z=z z P ¼ 1þ ¼ ¼ h h − ð Þ In Sec. III, the CE is then computed for the J 2 and r ð Þ 2 2 log 1 þ arctan : 3 P 3− f z z=zh zh J ¼ 2 nucleon families. Hence, two types of configura- tional-entropic Regge trajectories are interpolated, relating One expands the r coordinate at low temperature (zh → ∞) the CE of the nucleon resonances to both their radial and restrict, to the leading order and to the next leading excitation and their mass spectra. Therefore, the mass order, terms in the expansion of z in powers of r [47,48], spectra of the next generation of higher spin nucleon P ¼ 1þ P ¼ 3− 4 resonances, for both J 2 and J 2 nucleon fami- tr 8 r z ¼ r 1 − þ OðtrÞ ;tr ¼ : ð4Þ lies, are derived, from interpolation methods based on the 5 zh already detected nucleon resonances in LHC. The results are still compared to potential candidates in PDG [11]. The Hence, using the expansion in Eq. (4), the metric (1) can be finite temperature case is also investigated. Section IV is replaced by

106002-2 NUCLEONS AND HIGHER SPIN BARYON RESONANCES: AN … PHYS. REV. D 101, 106002 (2020) 2 3 5 dx⃗ One can perform a Kaluza-Klein expansion of the four- 2 ¼ 2AðrÞ = ð Þ 2 − 2 − ð Þ ds e fT r dt dr ; 5 dimensional transverse components of the field, fTðrÞ X ð Þ¼ ð Þ ð Þ¼ ΨL=Rð Þ¼ ΨL=Rð ÞτL=Rð Þ ð Þ where the warp factor reads A z log R=r and fT r x; r; T x n=2 r; T ; 13 4 4 1 − n r =zh. Considering the small correction to the dilaton, one defines a thermal dilaton as with ΨL=RðxÞ denoting the tower of Kaluza-Klein modes ϕ ð Þ¼ 2 2 ¼ 2 2ð1 þ ρ Þ ð Þ and n is the radial quantum number. Therefore, replacing T r kTr k r T ; 6 −3 ð Þ τL=Rð Þ¼ 2A r χL=Rð ÞðÞ where r; T e r; T 14 in the equation of motion obtained from the action (8) 9απ2 T2 T2 ρ ¼ þ δ þ δ þ OðT6Þ; ð7Þ yields a Schrödinger-type equation of motion for χðr; TÞ, T 16 T1 12 2 T2 12 2 F F given by with parameters, matching phenomenological data, fixed as ½−∂2 þ U ðr; TÞχL=Rðr; TÞ¼M2χL=Rðr; TÞ: ð15Þ α ¼ 0 δ ¼ − 8 δ ¼ − 4 ¼ 87 r L=R n in , T1 3, T2 9, and F MeV [48]. Ψ The action for fermionic field takes the following form The effective potential UL=Rðr; TÞ at finite temperature can at finite temperature: be split into two parts, Z pffiffiffi 4 ¯ ˆ UL=Rðr; TÞ¼ΛL=RðrÞþΩL=Rðr; TÞ; ð16Þ SB ¼ d xdr gΨðx; r; TÞDðrÞΨðx; r; TÞ; ð8Þ where where the Dirac operator reads 1 Λ ð Þ¼ 4 2 þ 2 2 l þ 1 ∓ i L=R r k r k Dˆ ð Þ¼ γM½∂ − ωab½γ γ − ½ ð Þþ ð Þ 2 r M M a; b m5 r; T VΨ r; T ; 2 ðl þ 1Þðl þ 1 1Þ ð Þ þ ; 9 r2 Ω ðr; TÞ¼2ρ k2ðk2r2Þ: ð17Þ for γa ¼ðγμ; −iγ5Þ denoting the Dirac matrices, γ5 ¼ L=R T γ0γ1γ2γ3 ϵa ¼ δa i is the volume element, M R M=z denotes The normalizable solutions of Eq. (15), for the baryons at ωab ¼ 1 ðδ½aδb Þ the tetrad, and M 4rf1=5ðrÞ r M denotes the spin finite temperature, read 3=10 connection. The quantity m5ðr; TÞ¼m5=f ðrÞ is the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! 2 2 lþ5 lþ3 temperature associated with the bulk mass in AdS L n −k r =2 2 lþ2 2 2 2 χ ðrÞ¼ e T k r Ln ðk r Þ; ð18Þ space. Here one considers m5 ¼ τ¯ − 2 [42], with τ¯ corre- n 5 T T Γðnþlþ 2Þ sponding to the twist dimension of the baryon operator at the boundary, which is related with the orbital angular sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! 2 2 lþ3 lþ1 momentum [40,49,50] as τ¯ ¼ l þ 3. For instance, the R n −k r =2 2 lþ1 2 2 2 χ ðrÞ¼ e T k r Ln ðk r Þ; ð19Þ P 1þ P 1− n Γð þlþ 3Þ T T J ¼ 2 states are related with l ¼ 0 and J ¼ 2 states n 2 with l ¼ 1. The dilaton temperature potential reads [47] where ΓðpÞ is the gamma function, Ln is the Laguerre ϕTðr; TÞ function, and the mass spectrum is specified by VΨðrÞ¼ : ð10Þ f3=10ðrÞ 3 2 ¼ 4 2 þ l þ ð Þ Mn kT n : 20 The baryon field can be split into left-handed, L, and right- 2 handed, R, components, The eigenfunctions (18), (19) are normalized according to Ψðx; zÞ¼ΨLðx; zÞþΨRðx; zÞ; ð11Þ Z ∞ L=R L=R where the chiral spinor components are given by dzχm ðr; TÞχn ðr; TÞ¼δmn: ð21Þ 0 1 ∓ γ5 In the zero temperature limit, T → 0, the solution of ΨL=Rðx; zÞ¼ Ψðx; zÞ: ð12Þ 2 Eq. (15) reads

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6 P 1þ 6×10 TABLE I. Mass spectra of the J ¼ 2 nucleon family reso- nances. The particle with an asterisk is left out the summary table 5×106 in PDG [11]. Experimental, JP =3/2– P – 6 AdS/QCD model, J =3/2 P 1þ 4×10 ¼ ) n Nucleon (J 2 ) Experimental AdS/QCD (soft wall) 2 JP ¼ 1þ nucleon mass spectra (MeV) 3×106

2 (MeV ð939Þ 939 49 0 05 2 0 N . . 1102.27 m 6 1 Nð1440Þ 1370 10 1423.02 2×10 2 Nð1710Þ 1700 20 1683.75 1×106 3 Nð1880Þ 1860 40 1909.19 4 Nð2100Þ 2100 50 2110.69 0 5 Nð2300Þ 2300þ40þ109 2294.56 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −30−0 n

P ¼ 3− sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FIG. 2. Mass spectra for nucleon resonances with J 2 3 obtained from the AdS/QCD model and the experimental 2n! − 2 2 2 lþ5 lþ2 lþ 2 2 χLð Þ¼ k r = 2 2ð Þ ð Þ n r 5 e k r Ln k r ; 22 values [11]. Γðn þ l þ 2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi realize that they match experimental values [11]. This is 2 ! 2 2 3 lþ1 n − 2 lþ lþ1 2 2 2 accomplished in Fig. 1. χRðrÞ¼ e k r = k 2r L ðk r Þ; ð23Þ n Γð þlþ 3Þ n P 3− n 2 Now, nucleon resonances with J ¼ 2 will be inves- tigated. They consist of excited states of nucleon particles, which correspond to the mass spectrum often corresponding to one of the quarks having a flipped spin state, or with different orbital angular momentum 3 when the particle decays. The parameter of the dilaton is 2 ¼ 4 2 þ l þ ð Þ Mn k n 2 : 24 fixed as k ¼ 0.47 GeV, having the best fit with the experimental results as one can see in Fig. 2. Table II It is worth to emphasize that in the infrared regime of the r shows the mass spectra in the soft wall model for baryons with JP ¼ 3− and their experimental results [11]. coordinate, the functions in Eq. (14) scale as τLðr; TÞ ∼ 2 9þl 7þl r2 and τRðr; TÞ ∼ r2 , whereas at the ultraviolet regime these functions vanish, exhibiting confinement [43]. III. CONFIGURATIONAL ENTROPY AND P 1þ SHANNON INFORMATION ENTROPY First, the nucleon family with J ¼ 2 will be approached. To obtain the best fit with the experimental When one takes into account some distribution of ¼ data at zero temperature, one fixes the parameter k probability pa, the CEP is defined, for discrete systems, P 1þ ð Þ 0.45 GeV for the J ¼ 2 nucleon resonances. For com- by (minus) the sum i pi log pi [3,5]. When N modes, parison, the experimental data from [11] are presented in constituting a discrete physical system, are described the third column and the soft wall result (20) in the fourth by a uniform distribution of probability, namely, ¼ 1 column of Table I. This makes possible to illustrate the pi =N, the CE has a maximal value equal to log N. P 1þ The differential CE regards the continuous limit of the CE mass spectra (20) of nucleon resonances with J ¼ 2 and [6,32]. The CE, underlying any physical system, has

6×106 the main pillar on the spatial correlations among the parts of the system. More precisely, the central structure of the 5×106 CE resides on the correlation of the fluctuations of some Experimental, JP =1/2+ ρðxÞ scalar field, which is a localized and Lebesgue AdS/QCD model, JP =1/2+ 4×106 integrable function describing the system. Given a vector ) 2 r ∈ Rd, the two-point correlation function is given by 3×106 (MeV 2 m 6 P 3− 2×10 TABLE II. Mass spectra of J ¼ 2 nucleon family resonances.

P 3− 1×106 n Nucleon (J ¼ 2 ) Experimental AdS/QCD (soft wall) P 3− J ¼ 2 nucleon mass spectra (MeV) 0 0 1 2 3 4 5 0 Nð1520Þ 1510 5 1486.27 n 1 Nð1700Þ 1700 50 1758.57 2 Nð1875Þ 1900 50 1994.04 P ¼ 1þ FIG. 1. Mass spectra of nucleon resonances with J 2 3 Nð2120Þ 2100 50 2204.49 obtained from the AdS/QCD model and experimental values [11].

106002-4 NUCLEONS AND HIGHER SPIN BARYON RESONANCES: AN … PHYS. REV. D 101, 106002 (2020) R GðrÞ¼ Rd ρðr˜ Þρðr þ r˜ Þdr˜ . The CE is nothing else than TABLE III. The CE for the nucleon family as a function of their the Shannon’s information theory involving correlations, if radial excitation n in the soft wall model. The second column P ¼ 1þ one establishes a distribution of probability using the two- displays the nucleon family with spin J 2 (and the higher point correlation function. To accomplish it, the so-called spin resonances N7, N8, N9), whereas the third column shows ðkÞ k their respective CE; the fourth column shows the nucleon family power spectrum, P , where is the wave vector, is P 3− with spin J ¼ 2 (and the higher spin resonances X5, X6, X7) defined to be the Fourier transform of the two-point and the fifth column illustrates their respective CE. correlation function [14]. The convolutionR and the ik·r 2 P ¼ 1þ P ¼ 3− Plancherel theorems yield PðkÞ ∼ k Rd ρðrÞe drk . nN(J 2 )CEN (J 2 )CE When one takes the Fourier transform 0 Nð939Þ 0.94981 Nð1520Þ 0.99604 Z 1 Nð1440Þ 1.06402 Nð1700Þ 1.14581 1 ð1710Þ ð1875Þ ρðkÞ¼ ρðrÞ −ik·r r ð Þ 2 N 1.15043 N 1.22903 p=2 e d ; 25 ð1880Þ ð2120Þ ð2πÞ Rd 3 N 1.21481 N 1.28816 4 Nð2100Þ 1.26504 X5 1.38194 ð2300Þ then the correlation distribution of probability, called as the 5 N 1.30562 X6 1.59038 6 N7 1.35258 X7 2.05683 modal fraction, reads [3] 7 N8 1.41702 8 N9 1.51519 jρðkÞj2 ϱðkÞ¼R ð Þ 2 d : 26 Rd jρðkÞj d k P 1þ For the J ¼ 2 nucleon family, the configurational- Therefore, the CE reads entropic Regge trajectory, relating log(CE) to the n quan- Z tum number, is the interpolation dashed line in Fig. 3. It has d CEρ ¼ − ϱ⋆ðkÞ log ϱ⋆ðkÞd k; ð27Þ the explicit expression Rd ð Þ¼1 06283 10−3 3 − 0 01712 2 ϱ ðkÞ¼ϱðkÞ ϱ ðkÞ CE n . × n . n where ⋆ = max . The expression of the CE is similar to the Gibbs entropy, which is defined for 0.13025n þ 0.94983: ð30Þ some statistical ensemble, of microstatesP having each − − ¼ Ei=kbT Ei=kbT We choose to use the third power in the interpolation, one probability pi e = i e , where Ei as it is sufficient to delimit within ∼0.17% the standard stands for the energy of the ith microstate and kb represents the Boltzmann constant. The interplay between the CE and deviation. statistical mechanics was paved in [6]. A second type of configurational-entropic Regge trajec- Both for the JP ¼ 1þ and the JP ¼ 3− nucleon families, tory, interpolating and relating the CE to (squared) mass 2 2 P ¼ 1þ given the Lagrangian L ruling the system in Eq. (8), the spectra of the nucleon family J 2 , is shown as the energy-momentum tensor is, in general, given by dashed curve in Fig. 4. The dashed curve in Figs. 3 and 4, respectively, pffiffiffi pffiffiffi ∂ð Þ ∂ð Þ correspond to Eqs. (30) and (31). For the plot in Fig. 4, μν 2 gL gL T ¼ pffiffiffi − ∂ α : ð28Þ x ∂gμν the configurational-entropic Regge trajectory has the fol- g ∂gμν ∂ð Þ ∂xα lowing form: ρð Þ ð Þ The z energy density corresponds to the T00 z com- 0.4 ponent of (28), emulating Eq. (47) of Ref. [23] for our specific case, as 0.3 M2 ρð Þ¼ ð Þ¼ n ½ðχLð ÞÞ2 þðχRð ÞÞ2 ð Þ r T00 r n r n r : 29 )) 0.2 n

r ( CE (

Hence, the energy density (25) in momentum space, the log 0.1 modal fraction (26), and the CE (27) can be numerically computed. 0.0 P 1þ The CE of the J ¼ 2 nucleons Nð939Þ, Nð1440Þ, ð1710Þ ð1880Þ ð2100Þ ð2300Þ N , N , N , N as a function of the –0.1 n quantum number is displayed in the third column of 0 1 2 3 4 5 Table III, for n ¼ 0; 1; …; 5. For higher values of n, the CE n P 1þ will be computed through the interpolation method, to be FIG. 3. log(CE) of the J ¼ 2 nucleons as a function of the n described in what follows. quantum number.

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0.4 Once the N9 nucleon resonance mass (34) has agreement to the mass 3122.9 1.6 of the Ξcð3123Þ baryon [11], with 0.3 P J and the still undetermined, the Ξcð3123Þ baryon might consist of a candidate to describe the N9 nucleon 0.2 0 ) resonance with mass (34). In the same way, the Ωcð3120Þ , CE ( with mass 3119.1 1.5 might also be described by the N9

log P 0.1 nucleon, if J and the isospin will be shown to match, indeed. ð1520Þ ð1700Þ ð1875Þ 0.0 Now, using the N , N , N , and Nð2120Þ nucleon family, the CE of each one of these P ¼ 3− –0.1 nucleons with J 2 is computed. Therefore, when the 1×106 2×106 3×106 4×106 5×106 CE is interpolated as a function of the n quantum numbers, 2 2 m (MeV ) P 3− a CE Regge trajectory is derived, for the J ¼ 2 nucleon P 1þ FIG. 4. The CE for the nucleon family J ¼ 2 information- family resonances, as a dashed curve in Fig. 5. The data in entropic Regge trajectory, with respect to the squared mass the fourth and fifth columns in Table III, for n ¼ 0,1,2,3, P 1þ spectra of the nucleon family J ¼ 2 . are better represented by the points in Fig. 5 as a function of n. −50 14 −14 6 The explicit expression of the CE Regge trajectory, log CEðmÞ¼3.09873 × 10 m − 1.16011 × 10 m represented by the interpolation dashed curve in Fig. 5, −7 2 1.42784 × 10 m − 0.167903; ð31Þ reads within 1.5% standard deviation. It is worth to emphasize CEðnÞ¼0.007780n3 − 0.058302n2 þ 0.190589n that in Eq. (36) m refers to the nucleon mass. − 0.003969: ð35Þ The mass spectra for the N7, N8, and N9 elements can be easily inferred by employing Eqs. (30), (31). In fact, for We opted to use a cubic interpolation, as it is sufficient to ¼ 6 ¼ 1 35258 n , Eq. (30) yields CE . . Then substituting delimit within ∼0.19% the standard deviation. this value in the CE Regge trajectory (31), after solving the A second type of configurational-entropic Regge trajec- subsequent equation yields the mass tory, relating the CE to (squared) mass spectra of the P 3− nucleon family J ¼ 2 , is shown in Fig. 6. m ¼ 2880 43 MeV: ð32Þ N7 The dashed curves in Figs. 5 and 6, respectively, correspond to Eqs. (35) and (36). For the plot in Fig. 6, It is worth to mention that this value of mass complies with the configurational-entropic Regge trajectory has the fol- Λ ð2940Þþ the order of the one for the c baryon, also with lowing form: P ¼ 3− 2939 6þ1.3 J 2 , with mass . −1.5 , detected in LHC [11]. P 1þ −20 6 −13 4 Similarly, for the next member N8 of the J ¼ 2 log CEðmÞ¼3.4700 × 10 m − 4.0410 × 10 m ¼ 7 nucleon family, substituting n into Eq. (30) implies 1.18206 × 10−7m2 þ 0.84686; ð36Þ that the corresponding CE has value 1.41702. Hence, replacing this value into the CE Regge trajectory (31) yields within 0.18% standard deviation. ¼ 3062 47 ð Þ mN8 MeV: 33 0.25

The N8 nucleon resonance matches the mass 3055.9 0.4 0.20 P of the Ξcð3055Þ baryon [11]. As other properties, like J Ξ ð3055Þ and the isospin are still undetermined for c , where 0.15 ) n only 894 events have been detected in LHC, the Ξcð3055Þ ( CE baryon might emulate a potential candidate to describe the 0.10 N8 nucleon resonance with mass (33), just in the case where P 3− 1 0.05 J ¼ 2 and the isospin 2 match the ones for Ξcð3055Þ. P 1þ Besides, the same can be accomplished for N9, J ¼ 2 0.00 nucleon, when Eq. (30) implies that CE ¼ 1.51519. N9 0.0 0.5 1.0 1.5 2.0 2.5 3.0 When replaced into Eq. (30), it produces the mass n P ¼ 3− ¼ 3169 51 ð Þ FIG. 5. CE of the J 2 nucleons as a function of the n mN9 MeV: 34 quantum number.

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1.2 0.25

1.0 0.20

0.8 0.15 ) )) m ( 0.6 GeV

0.10 ( CE ( m P + log N(J =1/2 ) 0.4 0.05 N(JP=3/2- )

0.2 0.00

6 6 6 6 6 6 2.0×10 2.5×10 3.0×10 3.5×10 4.0×10 4.5×10 0.0 2 2 m (MeV ) 0 50 100 150 T(MeV) P 3− FIG. 6. The CE for the nucleon family J ¼ 2 information- P ¼ 1þ P ¼ 3− entropic Regge trajectory, with respect to the squared mass FIG. 7. The masses of the ground state J 2 and J 2 as spectra a function of the temperature.

The mass spectra of the X5, X6, and X7 elements can be A. Finite temperature case easily inferred, by employing Eqs. (35), (36). In fact, for Applying the same procedure to compute the CE at n ¼ 4, Eq. (35) yields CE ¼ 1.3819. Then substituting this zero temperature, we evaluate the CE for lightest state of value in the CE Regge trajectory (36), and solving the baryons, as a function the temperature. The solutions of the resulting equation, the solution is the mass fields at finite temperature (22) and (23) are used. The baryons are dissociated at same temperature as can be seen in Fig. 7 ¼ 2232 11 ð Þ mX5 MeV: 37 using the relation (20). The same result is evidenced employ- P 1þ ing the CE. The results of CE for the ground states of J ¼ 2 This nucleon resonance matches the mass and other and JP ¼ 3− nucleon resonances at finite temperature are Ξð2250Þ 1 P 2 properties of the isospin 2 xi baryon, whose J presented in Fig. 8 as a function of the temperature T. From Ξð2250Þ still needs confirmation [11].However,the baryon Fig. 8, one can note that in the range between 0 and 36 MeV, is a potential candidate to represent the X5 nucleon P 1þ the CE is constant. In this range, for J ¼ 2 ,CE≈ 0.95,and resonance with mass (37). P 3− 1þ for J ¼ 2 , the CE equals 1.0. For higher temperatures, the Similarly, for the next member X6 of the spin nucleon 2 CE increases monotonically as a function of the temperature. n ¼ 5 family, substituting into Eq. (35) implies that the It suggests a decrement in the configurational stability of the corresponding CE has value 1.5903. Hence, replacing this nucleon resonances. The baryons are dissociated at same value into the CE Regge trajectory (36) yields temperature as can be seen in Fig. 7 using the relation (20). The same result is evidenced employing the CE. ¼ 2354 15 ð Þ mX6 MeV: 38 P 1þ P 3− For both the J ¼ 2 and J ¼ 2 nucleon families, a scaling law dependence can be implemented. First, Fig. 9 Analogous to the previously analyzed case consisting of the d 1þ X5 nucleon resonance with J ¼ 2 , also the X6 nucleon resonance does match the mass and other properties, like 3.0 the isospin of the Ξð2370Þ xi baryon [11]. The Ξð2370Þ baryon can be a potential candidate to describe the X6 2.5 nucleon resonance with mass (38). P 2.0 Besides, the same can be accomplished for the X7 J ¼ 1þ

¼ 2 0568 CE 2 nucleon, when Eq. (35) implies that CEX7 . . 1.5

When replaced into Eq. (35), it yields the mass N(JP=1/2+) 1.0 N(JP=3/2- ) ¼ 2473 26 ð Þ mX7 MeV: 39 0.5

Again, the derived mass (39) for the X7 nucleon resonance 0.0 P 3− 0 50 100 150 with J ¼ 2 matches the mass and other properties, like T(MeV) the isospin of the Ξð2500Þ xi baryon [11], which may consist of a potential candidate to emulate the X7 nucleon FIG. 8. The CE for the ground state of nucleon resonances P 1þ P 3− resonance with mass (39). J ¼ 2 and J ¼ 2 as a function of the temperature.

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both the zero temperature and finite temperature case. First, 3.0 we analyzed the zero temperature case using the CE obtained in the holographic model described in Sec. II.

2.5 Two types of configurational-entropic Regge trajectories were derived for each nucleon family, where the CE for

P + CE Fit- N(J =1/2 ) nucleon resonances was related to their quantum number n 2.0 N(JP=1/2+) and their mass spectra as well. Through this relation, the Fit- N(JP=3/2- ) next generation of nucleon families was inferred. In the 1.5 N(JP=3/2- ) P 1þ case of the J ¼ 2 nucleon family, the derived mass spectra, for the next generation of resonances in this 1.0 nucleon family corresponding to quantum numbers 0 50 100 150 ¼ 6 ¼ 2880 43 n , 7, 8, were, respectively, mN7 MeV, T(MeV) ¼ 3062 47 ¼ 3169 51 mN8 MeV. and mN9 MeV. On P 1þ P 3− P ¼ 3 2− FIG. 9. The CE for the nucleon families J ¼ 2 and J ¼ 2 the other hand, for the J = nucleon family, the information-entropic Regge trajectory at finite temperature, with derived mass spectra, for the next generation of resonances, respect to the mass spectra. corresponding to quantum numbers n ¼ 4, 5, 6, were ¼ 2232 11 ¼ 2354 15 mX5 MeV, mX6 MeV, and ¼ 2473 26 displays the numerical interpolation for nucleon resonances mX7 MeV. P 1þ The soft wall AdS/QCD model at small temperatures of J ¼ 2 , given by proposed in Refs. [47,48] for the description of fermionic −14 7 −11 6 CE1þ ðTÞ¼1.6289 × 10 T − 1.4016 × 10 T hadrons was employed to study the dissociation of nucleon 2 families in the context of CE. The CE of the ground state þ 4 7050 10−9 5 − 7 7737 10−7 4 P 1þ P 3− . × T . × T for the J ¼ 2 and J ¼ 2 nucleon families was analyzed þ 6.42109 × 10−5T3 − 2.27622 × 10−3T2 as a function of the temperature, as shown in Fig. (9). For both cases, the CE monotonically increases with the þ 0 0296115 þ 0 858299 ð Þ . T . : 40 temperature. It illustrates a decrement in the configurational stability of the system. The result obtained for the CE is Similarly, numerical interpolation for baryon resonances of consistent with the result analyzing the spectrum masses P ¼ 3− J 2 yields at finite temperature plotted in Fig. 7. One notices that

−14 7 −11 6 the mass decreases with increments in the temperature. CE3− ðTÞ¼3.21436 × 10 T − 2.46643 × 10 T 2 Besides, configurational-entropic Regge trajectories were þ 7.5303 × 10−9T5 − 1.1490 × 10−6T4 implemented, relating the CE with the temperature, for both P 1þ P 3− the J ¼ 2 and J ¼ 2 nucleon families. þ 8.8859 × 10−5T3 − 3.00518 × 10−3T2 þ 0.0377408T þ 0.88293; ð41Þ ACKNOWLEDGMENTS L. F. is grateful to the National Council for Scientific also evincing a scaling law for the CE as a function of and Technological Development—CNPq (Brazil) under the temperature. As the CE is constant in the range, Grant No. 153337/2018-4. R. d. R. is grateful to T ≲ 36 MeV, the scaling laws (40), (41) hold for T ≳ 60 MeV within less than 2% of accuracy. FAPESP (Grant No. 2017/18897-8), to the National Council for Scientific and Technological Development— IV. CONCLUDING REMARKS CNPq (Grants No. 303390/2019-0, No. 406134/2018-9, and No. 303293/2015-2), and to ICTP for partial financial In this paper, we computed the CE for the JP ¼ 1=2þ support. The authors thank Dr. Allan Gonçalves for fruitful and JP ¼ 3=2− nucleon families, in the soft wall model for discussions.

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